Glossary

Short definitions of the terms this book uses, from A to Z, each with a link to the chapter that teaches it.

A

AAS Two triangles are congruent if two angles and a side not between them match. It follows from ASA together with the exterior angle inequality, without using parallels. See Chapter 2.

Acute angle An angle between \(0°\) and \(90°\). See Chapter 1.

Acute triangle A triangle whose three angles are all acute. See Chapter 2.

Adjacent angles Two angles that share a vertex and an arm and do not overlap, like two slices of a cake next to each other. See Chapter 1.

Affine map A map \(\mathbf{x} \mapsto A\mathbf{x} + \mathbf{b}\) with \(A\) an invertible matrix. It keeps lines, parallels and ratios of areas, so any affine statement true for one triangle is true for all triangles. See Chapter 10.

Affine variety The set of common zeros of some polynomials, such as the circle \(x^2 + y^2 - 1 = 0\) or the twisted cubic in space. It is the basic object of algebraic geometry. See Chapter 21.

Alternate angles The two angles a transversal makes between two lines, on opposite sides of the transversal. They are equal exactly when the lines are parallel. See Chapter 3.

Altitude The perpendicular from a vertex of a triangle to the line of the opposite side; it can fall outside the triangle. The three altitudes meet at the orthocentre. See Chapter 2.

Altitude theorem In a right triangle, the altitude to the hypotenuse is the geometric mean of the two pieces it cuts the hypotenuse into. It gives a construction of \(\sqrt{a}\) with straightedge and compass. See Chapter 5.

AM-GM inequality For positive \(a\) and \(b\), the geometric mean is at most the arithmetic mean: \(\sqrt{ab} \le (a+b)/2\), with equality only when \(a = b\). A semicircle shows it in one picture. See Chapter 5.

Angle The figure made by two rays that share an endpoint, the vertex; the rays are its arms. It is written \(\angle AOB\) with the vertex in the middle. See Chapter 1.

Angle between vectors The angle \(\theta\) between two nonzero vectors drawn from the same tail, with \(0 \le \theta \le \pi\). It is found from the dot product: \(\cos\theta = \mathbf{u}\cdot\mathbf{v}/(|\mathbf{u}||\mathbf{v}|)\). See Chapter 9.

Angle bisector theorem The bisector of angle \(A\) in triangle \(ABC\) divides side \(BC\) in the ratio \(AB : AC\). With Ceva's theorem it shows that the three angle bisectors are concurrent. See Chapter 12.

Angle chasing Finding unknown angles in a figure step by step from the angle facts: angles on a line, vertical angles, parallel lines and the angle sum. See Chapter 3.

Angle defect (polyhedron) At a vertex of a polyhedron, \(360°\) minus the sum of the face angles there. Descartes found that the defects of a convex polyhedron always add up to \(720°\); on a surface it plays the role of the total curvature. See Chapter 11.

Angle of parallelism In hyperbolic geometry, the angle \(\Pi(d)\) between the perpendicular from a point to a line at distance \(d\) and a limiting parallel through the point. It satisfies \(\cos\Pi(d) = \tanh d\). See Chapter 17.

Angle sum of a triangle In Euclidean geometry the three angles of every triangle add to \(180°\). The proof needs the parallel postulate; on a sphere the sum is larger and in the hyperbolic plane smaller. See Chapter 3.

Antipodal points Two points of a sphere at opposite ends of a diameter. Two points that are not antipodal lie on exactly one great circle. See Chapter 16.

Antiprism A polyhedron made of two parallel copies of an \(n\)-gon, one turned by half a step, joined by a zigzag band of \(2n\) triangles. See Chapter 11.

Apollonius' problem Given three circles, draw a circle touching all three. In general there are eight solutions, and inversion turns the problem into finding a common tangent of two circles. See Chapter 13.

Apollonius' theorem In a triangle with median \(m_a\) to side \(a\), \(b^2 + c^2 = 2m_a^2 + \tfrac12 a^2\). It is the special case of Stewart's theorem for a median. See Chapter 12.

Arc A piece of a circle between two of its points. Two points cut a circle into a minor arc (the shorter) and a major arc. See Chapter 6.

Arc length The length of a curve \(\gamma\) from \(t = a\) to \(t = b\), \(\int_a^b |\gamma'(t)|\,dt\). It does not change under reparametrisation. See Chapter 19.

Archimedean solid A convex polyhedron whose faces are regular polygons of more than one kind, with every vertex alike. There are 13, apart from the prisms and antiprisms. See Chapter 11.

Archimedean tiling Another name for a semi-regular tiling: regular polygons of more than one kind, edge to edge, with the same polygons in the same cyclic order at every vertex. See Chapter 3.

Archimedes' axiom Hilbert's axiom V.1: given segments \(AB\) and \(CD\), some whole number of copies of \(CD\) laid along the ray \(AB\) goes past \(B\). No segment is infinitely small compared with another. See Chapter 15.

Area The amount of surface a figure covers, measured in unit squares. It is kept by moving the figure and adds when a figure is cut into pieces. See Chapter 4.

Argument (of a complex number) The angle \(\arg z\) from the positive real axis to the point \(z\). Every nonzero \(z\) is \(re^{i\theta}\) with \(r = |z|\) and \(\theta = \arg z\). See Chapter 13.

Arithmetic mean The number \((a+b)/2\) halfway between \(a\) and \(b\). See Chapter 5.

ASA Two triangles are congruent if two angles and the side between them match. It is proved from the SAS axiom. See Chapter 2.

Asymptotes The two lines \(y = \pm\frac{b}{a}x\) that the hyperbola \(x^2/a^2 - y^2/b^2 = 1\) approaches far from its centre. See Chapter 8.

Atlas A collection of charts that covers a manifold, like the pages of an atlas of the Earth. See Chapter 20.

Axiom A starting fact accepted without proof, also called a postulate. Every other fact of a geometry is proved from its axioms. See Chapter 1.

B

Basis A pair of vectors in the plane (three in space) such that every vector is exactly one linear combination of them. The coefficients are the coordinates in that basis. See Chapter 9.

Beltrami-Klein disc A model of the hyperbolic plane: points are the points strictly inside a circle and lines are its chords. Through a point not on a chord pass infinitely many chords that miss it, so the parallel postulate fails. See Chapter 15.

Binormal The unit vector \(B = T \times N\) of a space curve, perpendicular to its osculating plane. Its rate of turning measures the torsion. See Chapter 19.

Bisector A ray that splits an angle into two equal adjacent angles. See Chapter 1.

Bolyai-Gerwien theorem Two polygons can be cut into finitely many pieces and rearranged into each other (are scissors-congruent) if and only if they have the same area. Hilbert's third problem asks the same for polyhedra, where it fails. See Chapter 4.

Boundary at infinity In the Poincaré models, the circle or line that the hyperbolic plane approaches but does not include: the real axis with \(\infty\) for the upper half-plane, the unit circle for the disc. Its points are ideal points. See Chapter 17.

Brianchon's theorem The dual of Pascal's theorem: for a hexagon whose six sides touch a conic, the three lines joining opposite vertices are concurrent. See Chapter 18.

C

Cardioid The heart-shaped curve \(r = a(1 + \cos\theta)\) in polar coordinates, traced by a point on a circle rolling round another circle of the same size. See Chapter 8.

Cartan geometry Élie Cartan's joining of Klein's and Riemann's views: a space that looks, in each tiny region, like a Klein geometry, but may be curved, so that it need not have any symmetries at all. See Chapter 21.

Catalan solids The duals of the 13 Archimedean solids, such as the rhombic dodecahedron. Their faces are all alike but their vertices are not. See Chapter 11.

Cauchy-Schwarz inequality For any vectors, \(|\mathbf{u}\cdot\mathbf{v}| \le |\mathbf{u}||\mathbf{v}|\). It is what makes the angle formula meaningful, and it gives the triangle inequality for vectors. See Chapter 9.

Cavalieri's principle If two solids have equal cross-sectional areas at every height, they have equal volumes. It gives the volumes of slanted prisms, pyramids, cones and the sphere. See Chapter 11.

Central angle An angle with its vertex at the centre of a circle. It stands on the arc between its arms. See Chapter 6.

Central projection Drawing a scene as an eye sees it: each point is joined to the eye, and the picture is where the line meets the picture plane. It keeps lines but not lengths, angles or parallels. See Chapter 18.

Centroid The point where the three medians of a triangle meet, two thirds of the way along each. In vectors it is \(\tfrac13(A + B + C)\), the triangle's balance point. See Chapter 12.

Ceva's theorem Cevians \(AD\), \(BE\), \(CF\) of triangle \(ABC\) are concurrent exactly when the product of the signed ratios \(\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}\) is \(1\). It proves at once that medians, angle bisectors and altitudes are concurrent. See Chapter 12.

Cevian A line through a vertex of a triangle and a point on the line of the opposite side. Medians, altitudes and angle bisectors are cevians. See Chapter 12.

Chart A patch \(U\) of a manifold together with a homeomorphism \(\varphi\) from \(U\) onto an open set of \(\mathbb{R}^n\): it puts coordinates on the patch, like a page of an atlas. See Chapter 20.

Chord A segment joining two points of a circle. A chord through the centre is a diameter. See Chapter 6.

Christoffel symbols The coefficients \(\Gamma^k_{ij}\) that express the derivatives of the coordinate vectors in terms of themselves and the normal. They can be computed from the metric alone, which is the heart of the Theorema Egregium and of the geodesic equations. See Chapter 19.

Circle The set of all points of the plane at distance exactly \(r\) (the radius) from a fixed point (the centre). See Chapter 6.

Circular points at infinity Two complex points at infinity through which every circle passes. They are why two circles meet in at most 2 ordinary points: by Bézout's theorem they meet in 4, and the other 2 are always these. See Chapter 21.

Circumcentre The centre of the circumcircle of a triangle, where the perpendicular bisectors of the sides meet. It is equally far from the three vertices. See Chapter 12.

Circumcircle The unique circle through the three vertices of a triangle; its radius is written \(R\). See Chapter 12.

Co-interior angles The two angles a transversal makes between two lines on the same side of the transversal (also called same-side interior). They add to \(180°\) exactly when the lines are parallel. See Chapter 3.

Collinear Points are collinear if they lie on one line. See Chapter 1.

Common notions Euclid's general truths about quantities, such as "things equal to the same thing are equal to each other", stated before his five postulates. See Chapter 1.

Compass The construction tool that draws the circle with a constructed centre through another constructed point. See Chapter 14.

Complementary angles Two angles that add up to \(90°\). See Chapter 1.

Components (of a vector) The numbers that describe a vector in coordinates, such as \((3, 2)\) for "3 across and 2 up". See Chapter 9.

Concave A polygon is concave if some diagonal lies outside it: it has a dent. See Chapter 3.

Concurrent Lines are concurrent if they all pass through one point. See Chapter 12.

Cone A solid with a circular base and a curved surface joining it to one point, the apex: a pyramid with a circular base. Its volume is \(\tfrac13\pi r^2 h\). See Chapter 11.

Conformal A map is conformal if it keeps angles between curves. Inversions, Möbius transformations, stereographic projection and Mercator's map are conformal. See Chapter 13.

Congruent Two figures are congruent if one can be moved onto the other exactly, by sliding, turning and perhaps flipping; equivalently, if an isometry carries one to the other. See Chapter 2.

Congruent number A whole number that is the area of a right triangle with rational sides, such as 6 (the 3-4-5 triangle). Deciding which numbers are congruent is still an open problem. See Chapter 4.

Conic section A curve cut from a double cone by a plane: an ellipse, a parabola or a hyperbola. Each is the set of points whose distance to a focus is \(e\) times the distance to a directrix. See Chapter 8.

Conjugate (complex) The number \(\bar z = x - iy\), the mirror image of \(z = x + iy\) in the real axis. See Chapter 13.

Conjugation (of isometries) For isometries \(f\) and \(g\), the composite \(gfg^{-1}\): the motion \(f\) moved by \(g\). For example, conjugating a rotation about \(C\) by \(g\) gives a rotation by the same angle about \(g(C)\). See Chapter 10.

Constant curvature A Riemannian manifold has constant curvature \(k\) if every plane at every point has sectional curvature \(k\). The models are the sphere (\(k > 0\)), Euclidean space (\(k = 0\)) and hyperbolic space (\(k < 0\)). See Chapter 20.

Constructible number A real number that is a coordinate of a point reachable from \((0, 0)\) and \((1, 0)\) by finitely many straightedge-and-compass steps. The constructible numbers form a field, and each has degree a power of 2. See Chapter 14.

Contrapositive The statement "if not \(Q\) then not \(P\)" made from "if \(P\) then \(Q\)". It is always exactly as true as the original. See Chapter 1.

Converse The statement "if \(Q\) then \(P\)" made from "if \(P\) then \(Q\)". A true statement can have a false converse. See Chapter 1.

Convex A polygon is convex if every diagonal lies inside it; a solid is convex if the segment between any two of its points stays inside it. No dents, no tunnels. See Chapter 3.

Convex hull The smallest convex polygon containing a finite set of points, the shape a rubber band makes when it snaps round pins at the points. See Chapter 21.

Corresponding angles Two angles in the same position at the two crossings of a transversal with two lines. They are equal exactly when the lines are parallel. See Chapter 3.

Cosine In a right triangle, the adjacent side divided by the hypotenuse; in general, the \(x\)-coordinate of the point of the unit circle at angle \(\theta\). See Chapter 7.

Counterexample An example that satisfies the hypothesis of a statement but not its conclusion. One is enough to disprove the statement. See Chapter 1.

Cross product For vectors in space, the vector \(\mathbf{u} \times \mathbf{v}\) perpendicular to both, with length equal to the area of the parallelogram they span and direction given by the right-hand rule. See Chapter 9.

Cross-ratio For four points on a line, \((A, B; C, D) = \frac{CA}{CB}\Big/\frac{DA}{DB}\) with signed lengths. It is the one number that every projective transformation keeps; Möbius transformations keep the complex cross-ratio of four points of the plane. See Chapter 18.

Cross-ratio (complex) The value \((z, z_1; z_2, z_3) = \frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}\), the image of \(z\) under the Möbius map sending \(z_1, z_2, z_3\) to \(0, 1, \infty\). Möbius transformations keep it, and it is real exactly when the four points lie on one generalised circle. See Chapter 13.

Crystallographic restriction A rotation symmetry of a wallpaper pattern or a crystal lattice can only have order 1, 2, 3, 4 or 6. The proof: the trace of its matrix, \(2\cos\theta\), must be an integer. See Chapter 10.

Cuboctahedron The Archimedean solid with squares and triangles alternating, made by cutting each corner of a cube down to the midpoints of its edges; its vertex configuration is \(3.4.3.4\). See Chapter 11.

Curvature (of a curve) For a curve parametrised by arc length, \(\kappa = |T'| = |\gamma''|\): how fast the direction turns per unit length. A circle of radius \(r\) has curvature \(1/r\). See Chapter 19.

Cyclic group The group \(C_n\) of the \(n\) rotations about one centre by multiples of \(360°/n\), the symmetry group of a pinwheel. See Chapter 10.

Cyclic quadrilateral A quadrilateral whose four vertices lie on one circle. Its opposite angles add to \(180°\), and conversely. See Chapter 6.

Cylinder A solid with two parallel circular bases: a prism with a circular base. Its volume is \(\pi r^2 h\). See Chapter 11.

D

Defect (of a triangle) The amount \(180° - (\angle A + \angle B + \angle C)\) by which the angle sum falls short. In neutral geometry it is never negative and adds like an area; in hyperbolic geometry it is proportional to the area. See Chapter 15.

Degenerate conic A second-degree equation whose solutions are not a genuine curve: two lines, one line, a single point or nothing. See Chapter 8.

Degree (of an algebraic number) The degree of its minimal polynomial, the monic rational polynomial of least degree that it satisfies. A constructible number has degree a power of 2. See Chapter 14.

Degree (of an angle) The usual unit of angle: a full turn is \(360°\). One degree is 60 minutes and one minute is 60 seconds, a survival of Babylonian base 60. See Chapter 1.

Dehn invariant A quantity built from each edge length of a polyhedron times its dihedral angle, ignoring rational multiples of \(\pi\). It is unchanged by cutting and regluing, and the cube and the regular tetrahedron of equal volume have different ones. See Chapter 11.

Delaunay triangulation The triangulation of a set of sites that joins two sites whenever their Voronoi cells share an edge. It is the dual of the Voronoi diagram, and it avoids thin triangles. See Chapter 21.

Desargues' theorem If two triangles are in perspective from a point, they are in perspective from a line, and conversely. It holds in every projective plane that sits in a space of higher dimension, but not in the non-Desarguesian planes. See Chapter 18.

Descartes' circle theorem If four circles each touch the other three, their curvatures \(k_i = 1/r_i\) (negative for a circle enclosing the others) satisfy \((k_1 + k_2 + k_3 + k_4)^2 = 2(k_1^2 + k_2^2 + k_3^2 + k_4^2)\). See Chapter 13.

Determinant For two plane vectors \((a, c)\) and \((b, d)\), the number \(ad - bc\). It is the signed area of the parallelogram they span, and it is not zero exactly when they form a basis. See Chapter 9.

Developable surface A surface with Gaussian curvature \(K = 0\) everywhere, such as a cylinder or a cone. It can be unrolled flat without stretching, which a sphere never can. See Chapter 19.

Diagonal A segment joining two vertices of a polygon that are not neighbours. See Chapter 3.

Diameter A chord through the centre of a circle; its length is twice the radius. See Chapter 6.

Differential (of a map) For a smooth map \(F : M \to N\) of manifolds, the linear map \(dF_p : T_pM \to T_{F(p)}N\) that sends the velocity of a curve to the velocity of its image. See Chapter 20.

Dihedral angle The angle between the two faces of a polyhedron that meet along an edge. See Chapter 11.

Dihedral group The group \(D_n\) of \(2n\) symmetries of a regular \(n\)-gon: \(n\) rotations and \(n\) reflections in mirrors through the centre. By Leonardo's theorem every finite symmetry group of a plane figure is cyclic or dihedral. See Chapter 10.

Dilation The map with centre \(O\) and scale factor \(k > 0\) that sends each point \(P\) to the point \(P'\) on the ray \(OP\) with \(OP' = k \cdot OP\). It multiplies all distances by \(k\) and sends lines to parallel lines; also called a homothety. See Chapter 5.

Direct isometry An isometry that keeps the orientation (the clockwise order of the corners of every triangle): a translation or a rotation. See Chapter 10.

Directed angle The angle \(\measuredangle(\ell, m)\) through which line \(\ell\) must turn anticlockwise to be parallel to line \(m\), measured modulo \(180°\). It removes the case splits in angle chasing with circles. See Chapter 12.

Directrix The fixed line \(\ell\) in the focus-directrix definition of a conic: the points whose distance to the focus is \(e\) times their distance to \(\ell\). See Chapter 8.

Disc A circle together with its inside. See Chapter 6.

Distance formula The distance between \((x_1, y_1)\) and \((x_2, y_2)\) is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), Pythagoras in coordinates. See Chapter 8.

Distance from a point to a line The length of the perpendicular from the point to the line, the shortest segment from the point to the line. See Chapter 3.

Dot product The number \(\mathbf{u}\cdot\mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\), also called the scalar product. It equals \(|\mathbf{u}||\mathbf{v}|\cos\theta\), so it gives angles and tests for perpendicularity. See Chapter 9.

Doubling the cube The Greek problem of constructing the side of a cube with twice the volume of a given one, that is \(\sqrt[3]{2}\). It is impossible with straightedge and compass, because \(\sqrt[3]{2}\) has degree 3. See Chapter 14.

Dual polyhedron The polyhedron with a vertex for each face of the old one and a face for each old vertex. The cube and octahedron are dual, as are the dodecahedron and icosahedron; the tetrahedron is its own dual. See Chapter 11.

Duality (projective) In the projective plane, any theorem about points, lines and incidence stays true when the words "point" and "line" are swapped. Pascal's theorem is dual to Brianchon's. See Chapter 18.

E

Eccentricity The number \(e > 0\) in the focus-directrix definition of a conic: an ellipse if \(e < 1\), a parabola if \(e = 1\) and a hyperbola if \(e > 1\). The orbit of Mars has \(e \approx 0.0934\). See Chapter 8.

Edge A segment where two faces of a polyhedron meet. See Chapter 11.

Ellipse The set of points whose distances to two foci add up to a constant \(2a\); in standard position \(x^2/a^2 + y^2/b^2 = 1\). It is the conic with eccentricity less than 1, and the shape of every planet's orbit. See Chapter 8.

Elliptic geometry The geometry of the elliptic plane, whose points are pairs of antipodal points of a sphere and whose lines are great circles. Two points lie on one line, two lines meet in one point, and there are no parallels. See Chapter 16.

Elliptic point A point of a surface where the Gaussian curvature is positive: the surface bends the same way in every direction, like a dome. See Chapter 19.

Endpoints The two points at the ends of a segment, or the starting point of a ray. See Chapter 1.

Equation of a plane The equation \(ax + by + cz = d\) of a plane in space, where \((a, b, c)\) is a normal vector. See Chapter 9.

Equidistant curve In the hyperbolic plane, the set of points at a fixed distance from a line. It is not a hyperbolic line; in the disc model it is an arc meeting the boundary at the line's endpoints. See Chapter 17.

Equilateral triangle A triangle with all three sides equal; its angles are all \(60°\). See Chapter 2.

Erlangen programme Felix Klein's idea (1872) that a geometry is the study of what a group of transformations keeps. Euclidean, similarity, affine and projective geometry form a ladder of larger and larger groups. See Chapter 21.

Euclidean group The group of all isometries of the plane under composition. Every element is \(\mathbf{x} \mapsto A\mathbf{x} + \mathbf{b}\) with \(A\) an orthogonal matrix. See Chapter 10.

Euclid's formula Every primitive Pythagorean triple is \((m^2 - n^2, 2mn, m^2 + n^2)\) for coprime \(m > n > 0\) of opposite parity, with the two legs perhaps swapped. See Chapter 4.

Euclid's postulates The five starting facts of the Elements: a line through two points, extending a segment, a circle with any centre and radius, all right angles equal, and the parallel postulate. See Chapter 1.

Euler characteristic The number \(\chi = V - E + F\). It is 2 for every convex polyhedron and every sphere-like surface, 0 for the torus and \(2 - 2g\) for a surface with \(g\) holes; it is the first invariant of topology. See Chapter 11.

Euler line The line through the circumcentre \(O\), centroid \(G\) and orthocentre \(H\) of any triangle, with \(HG = 2GO\). The nine-point centre lies on it too, halfway between \(O\) and \(H\). See Chapter 12.

Euler's formula (polyhedra) For every convex polyhedron, \(V - E + F = 2\), where \(V\), \(E\), \(F\) are the numbers of vertices, edges and faces. See Chapter 11.

Euler's inequality For any triangle, the circumradius is at least twice the inradius, \(R \ge 2r\), with equality only for the equilateral triangle. See Chapter 12.

Excircle A circle outside a triangle touching one side and the extensions of the other two. Its centre, the excentre, is where one internal and two external angle bisectors meet. See Chapter 12.

Exponential map The map \(\exp_p\) that sends a tangent vector \(v\) at \(p\) to the point reached by following the geodesic with initial velocity \(v\) for time 1. See Chapter 20.

Extended law of sines In any triangle, \(a/\sin A = b/\sin B = c/\sin C = 2R\), where \(R\) is the circumradius. See Chapter 12.

Extended plane The complex plane with one point at infinity added, \(\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}\), on which Möbius transformations act. It is the Riemann sphere. See Chapter 13.

Exterior angle inequality An exterior angle of a triangle is greater than either remote interior angle. It is proved without parallels and so holds in neutral geometry. See Chapter 2.

Exterior angle (of a polygon) At a vertex, the angle between one side and the extension of the next: \(180°\) minus the interior angle. For any convex polygon the exterior angles add up to \(360°\). See Chapter 3.

Exterior angle (of a triangle) The angle at a vertex between one side and the extension of the other side beyond the vertex. It equals the sum of the two remote interior angles. See Chapter 2.

F

Face One of the flat polygons that bound a polyhedron. See Chapter 11.

Faltings's theorem A curve of genus at least 2 defined by rational polynomials has only finitely many rational points (Faltings 1983, conjectured by Mordell). The shape of the curve governs its arithmetic. See Chapter 21.

Fano plane The smallest projective plane: seven points and seven lines, three points on each line and three lines through each point, with no two lines parallel. Its diagonal points are collinear, which never happens in the real plane. See Chapter 18.

Fermat prime A prime of the form \(2^{2^m} + 1\). Only five are known: \(3\), \(5\), \(17\), \(257\) and \(65537\). See Chapter 14.

Feuerbach's theorem The nine-point circle of a triangle touches the incircle from inside and the three excircles from outside (Feuerbach 1822). See Chapter 12.

Fibonacci numbers The sequence \(1, 1, 2, 3, 5, 8, 13, \ldots\), each the sum of the two before. Ratios of neighbours tend to the golden ratio. See Chapter 5.

First fundamental form The quadratic form \(I = E\,du^2 + 2F\,du\,dv + G\,dv^2\) (also written \(ds^2\)) of a surface, made from the dot products of the coordinate vectors. It gives lengths, angles and areas measured inside the surface. See Chapter 19.

Flat torus The square with opposite sides glued and the flat metric \(ds^2 = dx^2 + dy^2\). It has curvature zero everywhere, and cannot be built in ordinary 3-space with its lengths kept. See Chapter 20.

Focus A fixed point in the definition of a conic. An ellipse or hyperbola has two foci; a parabolic mirror sends light from its focus out in parallel rays. See Chapter 8.

Frenet frame The three perpendicular unit vectors \(T\) (tangent), \(N\) (principal normal) and \(B\) (binormal) attached to each point of a space curve. The Frenet-Serret equations say how it turns, in terms of curvature and torsion. See Chapter 19.

Frieze group The symmetry group of a frieze, a pattern along a strip that repeats. There are exactly seven. See Chapter 10.

G

Gauss-Bonnet theorem For every compact surface \(S\) without boundary, \(\iint_S K\,dA = 2\pi\chi(S)\): the total curvature depends only on the topology. For a geodesic triangle, the angle sum minus \(\pi\) is the curvature inside it. See Chapter 19.

Gaussian curvature The product \(K = k_1 k_2\) of the principal curvatures of a surface at a point. It is \(1/R^2\) on a sphere, \(0\) on a plane or cylinder, and negative at a saddle. See Chapter 19.

Gauss-Wantzel theorem A regular \(n\)-gon can be constructed with straightedge and compass exactly when \(n = 2^k p_1 \cdots p_r\) with the \(p_i\) distinct Fermat primes. So the 17-gon can be constructed and the 7-gon cannot. See Chapter 14.

General position Four points of the projective plane are in general position if no three of them lie on a line. Any four such points can be moved to any other four by a projective transformation. See Chapter 18.

Generalised circle A circle or a line, a line being a circle through the point at infinity. Inversions and Möbius transformations send generalised circles to generalised circles. See Chapter 13.

Geodesic A curve on a surface with geodesic curvature zero: its acceleration is purely normal to the surface. Geodesics are the straightest curves, and short pieces of them are the shortest paths; on a sphere they are great circles. See Chapter 19.

Geodesic curvature The part of a surface curve's acceleration that lies in the tangent plane, with a sign: the turning that an ant walking along the curve feels. See Chapter 19.

Geodesic (Riemannian) A curve \(x(t)\) in a Riemannian manifold satisfying the geodesic equations \(\ddot x^k + \Gamma^k_{ij}\dot x^i \dot x^j = 0\). A point and a velocity determine one geodesic. See Chapter 20.

Geometric mean The number \(\sqrt{ab}\), the side of the square with the same area as the rectangle \(a \times b\). The Greeks called it the mean proportional between \(a\) and \(b\). See Chapter 5.

Geometrization conjecture Thurston's statement that every closed orientable 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Perelman proved it, and with it the Poincaré conjecture, in 2002-2003. See Chapter 21.

Gergonne point The point where the three cevians from the vertices of a triangle to the points of contact of the incircle meet. Ceva's theorem shows they are concurrent. See Chapter 12.

Girard's theorem A spherical triangle with angles \(\alpha\), \(\beta\), \(\gamma\) on a sphere of radius \(R\) has area \((\alpha + \beta + \gamma - \pi)R^2\): area is spherical excess. See Chapter 16.

Glide reflection A reflection in a line followed by a translation by a nonzero vector parallel to that line, like a row of footprints. See Chapter 10.

Golden ratio The number \(\varphi = (1 + \sqrt5)/2 \approx 1.618\) that comes from cutting a segment in extreme and mean ratio. It is the ratio of a regular pentagon's diagonal to its side. See Chapter 5.

Golden rectangle A rectangle with sides in ratio \(\varphi : 1\). Cutting off a square leaves a smaller golden rectangle. See Chapter 5.

Gram-Schmidt process The method of turning any basis into an orthonormal one: subtract from each vector its projections on the earlier ones, then scale to length 1. See Chapter 9.

Great circle The circle cut from a sphere by a plane through its centre, like the equator or a meridian. Great circles are the lines of spherical geometry, and arcs of them are the shortest paths on the sphere. See Chapter 16.

H

Hairy ball theorem Every continuous vector field on the sphere \(S^2\) is zero somewhere: you cannot comb a hairy ball flat. See Chapter 20.

Half-turn A rotation by \(180°\). It sends each point \(P\) to the point \(P'\) with the centre the midpoint of \(PP'\). See Chapter 10.

Harmonic conjugate When the cross-ratio \((A, B; C, D) = -1\), \(D\) is the harmonic conjugate of \(C\) with respect to \(A\) and \(B\): \(C\) divides \(AB\) inside in some ratio and \(D\) divides it outside in the same ratio. See Chapter 18.

Haversine formula A formula for the great-circle distance between two points from their latitudes and longitudes, using \(\operatorname{hav}\theta = \sin^2(\theta/2)\). It stays accurate for short distances, where the spherical law of cosines loses digits. See Chapter 16.

Heronian triangle A triangle whose sides and area are all whole numbers, such as the 13-14-15 triangle with area 84. See Chapter 4.

Heron's formula A triangle with sides \(a\), \(b\), \(c\) and semi-perimeter \(s = \tfrac12(a + b + c)\) has area \(\sqrt{s(s-a)(s-b)(s-c)}\). See Chapter 4.

Heron's shortest path The shortest path from a point to a line and on to another point on the same side bounces off the line with equal angles: the law of reflection, found by reflecting one point in the line. See Chapter 2.

Hilbert's axioms David Hilbert's axioms for Euclidean geometry (1899), in five groups: incidence, order, congruence, parallels and continuity. They fill the gaps in Euclid, such as the missing betweenness facts. See Chapter 15.

Hilbert's third problem Can any two polyhedra of equal volume be cut into finitely many pieces that reassemble from one into the other? Dehn answered no in 1900; the cube and the regular tetrahedron are a counterexample. See Chapter 11.

Hinge theorem If two triangles have two sides equal in pairs, the one with the larger angle between those sides has the larger third side. See Chapter 2.

Homeomorphism A continuous map with a continuous inverse. Two spaces related by a homeomorphism are the same to a topologist. See Chapter 20.

Homogeneous coordinates Coordinates \([x : y : z]\) for a point of the projective plane, the coordinates of any nonzero point on the corresponding line through the origin. They are defined only up to a common nonzero factor; \(z = 0\) gives the points at infinity. See Chapter 18.

Homothety Another name for a dilation: the map \(P \mapsto C + k\,\overrightarrow{CP}\) with centre \(C\) and ratio \(k\). See Chapter 10.

Horizon (in perspective) The line in a perspective picture where the vanishing points of all horizontal directions lie: the image of the line at infinity of the ground. See Chapter 18.

Hyperbola The set of points whose distances to two foci differ by a constant; in standard position \(x^2/a^2 - y^2/b^2 = 1\). It is the conic with eccentricity greater than 1, and it has two branches and two asymptotes. See Chapter 8.

Hyperbolic geometry The geometry in which, through a point not on a line, there are at least two (and then infinitely many) parallels. Triangles have angle sum less than \(180°\), and the area of a triangle is its defect. See Chapter 17.

Hyperbolic parallel axiom Through a point not on a line there are at least two lines that do not meet it. Replacing Playfair's axiom by it gives hyperbolic geometry. See Chapter 15.

Hyperbolic point A point of a surface where the Gaussian curvature is negative: the principal curvatures have opposite signs and the surface crosses its tangent plane, like a saddle. See Chapter 19.

Hyperboloid model The model of the hyperbolic plane as the upper sheet of \(x_1^2 + x_2^2 - x_3^2 = -1\), with distances from the Lorentz form. It is the space of velocities of special relativity. See Chapter 17.

Hypotenuse The side of a right triangle opposite the right angle, its longest side. See Chapter 2.

Hypothesis The part of a theorem that says what may be assumed. The conclusion says what must be proved. See Chapter 1.

I

Ideal triangle A hyperbolic triangle whose three vertices are on the boundary at infinity. All its angles are 0, so its area is \(\pi\), the largest any hyperbolic triangle can have. See Chapter 17.

If and only if "\(P\) if and only if \(Q\)", written \(P \iff Q\), means that \(P\) implies \(Q\) and \(Q\) implies \(P\): a statement and its converse are both true. See Chapter 1.

Image The point \(f(P)\) that a transformation \(f\) sends \(P\) to. See Chapter 10.

Incentre The centre of the incircle of a triangle, where the three angle bisectors meet. It is equally far from the three sides. See Chapter 12.

Incidence Which points lie on which lines. It is what central projection keeps, and the subject of projective geometry. See Chapter 18.

Incircle The circle inside a triangle touching all three sides; its radius is written \(r\) and the area is \(rs\), with \(s\) the semiperimeter. See Chapter 12.

Incommensurable Two lengths are incommensurable if no common unit measures both a whole number of times, like the side and diagonal of a square. See Chapter 5.

Independent (axiom) A statement is independent of a set of axioms if neither it nor its negation can be proved from them. A model in which the axioms hold and the statement fails shows it cannot be proved. See Chapter 15.

Inscribed angle An angle whose vertex is on a circle and whose arms are chords. It stands on (subtends) the arc between its arms. See Chapter 6.

Inscribed angle theorem An inscribed angle is half the central angle on the same arc. So all inscribed angles on the same arc are equal. See Chapter 6.

Inscribed square problem Toeplitz's question (1911): does every Jordan curve contain four points that are the corners of a square? It is proved for smooth curves but open in general. See Chapter 21.

Intercept theorem A line parallel to one side of a triangle cuts the other two sides in the same ratio, and conversely. It is the key to similar triangles. See Chapter 5.

Intrinsic A property of a surface is intrinsic if it depends only on the first fundamental form, that is, on measurements made inside the surface. Length, angle, area and (by the Theorema Egregium) Gaussian curvature are intrinsic. See Chapter 19.

Inversion The map in a circle with centre \(O\) and radius \(r\) that sends \(P\) to the point \(P'\) on the ray \(OP\) with \(OP \cdot OP' = r^2\). It fixes the circle, swaps inside and outside, sends generalised circles to generalised circles and keeps angles. See Chapter 13.

Isometry A transformation that keeps all distances. In the plane every isometry is a translation, rotation, reflection or glide reflection. See Chapter 10.

Isosceles triangle A triangle with at least two equal sides. The equal sides are the legs, the third is the base, and the angles at the base are the base angles. See Chapter 2.

Isosceles triangle theorem In an isosceles triangle the base angles are equal (Euclid's pons asinorum); conversely, a triangle with two equal angles is isosceles. See Chapter 2.

J

Jordan curve theorem A closed curve in the plane that does not cross itself divides the plane into exactly two regions, an inside and an outside. It is obvious in pictures and hard to prove. See Chapter 11.

K

Kepler conjecture No packing of equal balls in space is denser than the cannonball packing, which fills \(\pi/\sqrt{18} \approx 0.7405\) of space. Hales proved it in 1998 with a computer; the Flyspeck project checked the proof formally in 2014. See Chapter 21.

Kepler's laws Each planet moves on an ellipse with the Sun at one focus; the line from the Sun sweeps out equal areas in equal times; and the square of the period is proportional to the cube of the semi-major axis. See Chapter 8.

Kissing number The largest number of unit balls that can touch one central unit ball: 2 on a line, 6 in the plane and 12 in space. See Chapter 21.

Kite A quadrilateral with two pairs of equal adjacent sides. Its diagonals are perpendicular, and one of them bisects the other. See Chapter 2.

L

Lattice The set of translations of a wallpaper pattern: all whole-number combinations \(m\mathbf{a} + n\mathbf{b}\) of two shortest independent translations. See Chapter 10.

Law of cosines In any triangle, \(c^2 = a^2 + b^2 - 2ab\cos C\). It is Pythagoras with a correction term, and it solves a triangle from three sides or from two sides and the angle between them. See Chapter 7.

Law of reflection A ray of light leaves a mirror at the same angle as it arrives. It follows from Heron's shortest path: light takes the shortest route. See Chapter 2.

Law of sines In any triangle, \(a/\sin A = b/\sin B = c/\sin C\). See Chapter 7.

Legs The two sides of a right triangle that form the right angle, or the two equal sides of an isosceles triangle. See Chapter 2.

Length (of a vector) The length of any arrow representing the vector, \(|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\); also called its magnitude or norm. See Chapter 9.

Leonardo's theorem Every finite group of isometries of the plane fixes a point (the average of an orbit), and so is a cyclic group \(C_n\) or a dihedral group \(D_n\). See Chapter 10.

Limiting parallels In the hyperbolic plane, the two lines through a point that approach a given line at its two ends without meeting it. They separate the lines that meet the given line from the ultraparallels. See Chapter 17.

Limiting points For two circles that do not cross, the two points that are inverse to each other in both circles. Inverting about one of them makes the circles concentric, which proves Steiner's porism. See Chapter 13.

Line A straight figure with no thickness that goes on forever in both directions. Through any two different points there is exactly one line. See Chapter 1.

Line at infinity The line made of all the points at infinity of the plane, one for each family of parallel lines. In a perspective drawing it becomes the horizon. See Chapter 18.

Linear combination A sum of scalar multiples of vectors, such as \(s\mathbf{u} + t\mathbf{v}\). See Chapter 9.

Linear pair Two adjacent angles whose outer arms are opposite rays, so that together they make a straight angle. The two angles of a linear pair add to \(180°\). See Chapter 1.

Locally isometric Two surfaces are locally isometric if they have charts with the same first fundamental form: an ant could not tell them apart by measuring inside them. A plane and a cylinder are locally isometric. See Chapter 19.

Locus The set of all points with some property. The perpendicular bisector of \(AB\) is the locus of points equally far from \(A\) and \(B\). See Chapter 2.

Lorentz form The form \(\langle \mathbf{x}, \mathbf{y} \rangle = x_1 y_1 + x_2 y_2 - x_3 y_3\) on \(\mathbb{R}^3\), with one minus sign. It is the geometry of Minkowski spacetime, and it gives the distances of the hyperboloid model. See Chapter 17.

Lorentzian metric A metric like a Riemannian one but with signature \((-, +, +, +)\), one minus sign. The spacetime of relativity carries one. See Chapter 20.

Loxodromic transformation A Möbius transformation that is neither parabolic, elliptic nor hyperbolic: it is conjugate to a spiral \(z \mapsto kz\) with \(k\) not real and \(|k| \ne 1\). See Chapter 13.

Lune The crescent-shaped region between two great semicircles meeting at angle \(\alpha\). On a sphere of radius \(R\) its area is \(2\alpha R^2\). See Chapter 16.

M

Major arc The longer of the two arcs into which two points cut a circle. See Chapter 6.

Manifold A space that looks near each point like ordinary \(n\)-dimensional space: every point has a neighbourhood with a homeomorphism to an open set of \(\mathbb{R}^n\). Surfaces are 2-manifolds. See Chapter 20.

Marked ruler A ruler with two marks on it, used for neusis (verging): sliding a segment of given length between two curves. With it Archimedes trisected any angle. See Chapter 14.

Mean curvature The average \(H = \tfrac12(k_1 + k_2)\) of the principal curvatures of a surface. Surfaces with \(H = 0\) are minimal surfaces, like soap films. See Chapter 19.

Medial triangle The triangle whose corners are the midpoints of the sides of a triangle. It is the image of the triangle under the dilation with centre the centroid and factor \(-\tfrac12\). See Chapter 12.

Median The segment from a vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid. See Chapter 2.

Menelaus' theorem Points \(D\), \(E\), \(F\) on the lines of the sides \(BC\), \(CA\), \(AB\) of a triangle are collinear exactly when the product of signed ratios \(\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}\) is \(-1\). See Chapter 12.

Mercator projection The cylindrical map of the sphere (Mercator 1569) that stretches north-south by the same factor \(\sec\varphi\) as east-west. It is conformal, so constant compass bearings are straight lines, but it greatly enlarges areas near the poles. See Chapter 16.

Method of exhaustion The Greek way of finding areas and volumes by squeezing a figure between inscribed and circumscribed shapes whose sizes are known, with a careful limiting argument. See Chapter 11.

Midline theorem The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. See Chapter 9.

Midpoint The point \(M\) on segment \(AB\) with \(AM = MB\). In coordinates it is the average \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). See Chapter 1.

Minimal polynomial The monic rational polynomial of least degree with a given algebraic number as a root. It is irreducible, and its degree is the degree of the number. See Chapter 14.

Minimal surface A surface with mean curvature \(H = 0\) everywhere. Such surfaces are exactly the critical points of area, like a soap film on a wire frame. See Chapter 19.

Minor arc The shorter of the two arcs into which two points cut a circle. See Chapter 6.

Möbius transformation A map \(f(z) = \frac{az + b}{cz + d}\) of the extended plane with \(ad - bc \ne 0\). It is a composite of inversions and isometries, it sends generalised circles to generalised circles, and exactly one sends any three points to any other three. See Chapter 13.

Model geometry A simply connected 3-manifold with a metric whose isometry group is as large as possible, acting transitively. Thurston showed there are exactly eight, including the spherical, Euclidean and hyperbolic ones. See Chapter 21.

Model (of axioms) An interpretation of the undefined terms (what counts as a point, a line, lying on) in which all the axioms are true. A model in which the axioms hold and a statement fails shows that the statement is independent of them. See Chapter 15.

Moduli space The space whose points are the different shapes of one kind of geometric object, such as the flat metrics on a torus up to scale. For tori it is 2-dimensional; for a surface of genus \(g \ge 2\) it has dimension \(6g - 6\). See Chapter 20.

Modulus The distance \(|z| = \sqrt{x^2 + y^2}\) of the complex number \(z = x + iy\) from \(0\). The distance between points \(z\) and \(w\) is \(|z - w|\). See Chapter 13.

Mohr-Mascheroni theorem Every point that can be constructed with straightedge and compass can be constructed with the compass alone. See Chapter 14.

Morley's theorem In any triangle, the three points where the angle trisectors next to each side meet form an equilateral triangle. See Chapter 12.

Mostow's rigidity theorem A closed (or finite-volume) hyperbolic manifold of dimension at least 3 has only one hyperbolic metric: its shape is determined by its topology. So its volume is a topological invariant. See Chapter 21.

Moving sofa problem What is the largest area of a shape that can be moved round a right-angled corner in a corridor of width 1? Gerver's sofa, with area about \(2.2195\), is conjectured to be the answer. See Chapter 21.

N

Napoleon's theorem If equilateral triangles are built outward on the three sides of any triangle, their centres form an equilateral triangle. Complex numbers give a short proof. See Chapter 13.

Net A flat piece of polygons, joined along edges, that folds up into a polyhedron. See Chapter 11.

Neutral geometry The geometry of all the theorems that follow from Hilbert's axioms without the parallel axiom. Its theorems hold in both Euclidean and hyperbolic geometry; it includes SAS, ASA, the exterior angle inequality and the Saccheri-Legendre theorem. See Chapter 15.

Nine-point circle The circle through the three midpoints of the sides of a triangle, the three feet of the altitudes and the three midpoints between the orthocentre and the vertices. Its centre is halfway between the circumcentre and orthocentre, and its radius is \(R/2\). See Chapter 12.

Non-Desarguesian plane A projective plane in which Desargues' theorem fails. Such planes exist, so Desargues' theorem cannot be proved from the plane incidence axioms alone. See Chapter 18.

Non-orientable A surface is non-orientable if a narrow band carried once around some loop comes back with its sides swapped, as in a Möbius strip. The real projective plane (and the elliptic plane) is non-orientable. See Chapter 18.

Normal curvature The signed curvature, at a point of a surface, of the normal section in a given tangent direction: the curve cut from the surface by the plane containing that direction and the normal. See Chapter 19.

Normal vector A vector perpendicular to a plane. A plane is fixed by one point on it and a normal vector. See Chapter 9.

O

Obtuse angle An angle between \(90°\) and \(180°\). See Chapter 1.

Obtuse triangle A triangle with one obtuse angle. See Chapter 2.

Opposite isometry An isometry that reverses orientation: a reflection or a glide reflection. See Chapter 10.

Opposite rays Two rays \(AB\) and \(AC\) from the same point going in opposite directions along one line. Together they make the whole line. See Chapter 1.

Orbifold A space like a manifold but allowed to have special points such as cone points and mirror edges. John Conway's orbifold notation classifies the 17 wallpaper groups by the orbifold each pattern folds up into. See Chapter 10.

Orbit The set of all images \(g(P)\) of a point \(P\) under a group of transformations. See Chapter 10.

Orthocentre The point where the three altitudes of a triangle meet. See Chapter 12.

Orthogonal Two vectors are orthogonal (perpendicular) when their dot product is zero. See Chapter 9.

Orthogonal matrix A matrix whose columns are perpendicular unit vectors. The linear isometries of the plane are given by orthogonal matrices: rotations have determinant \(1\) and reflections \(-1\). See Chapter 10.

Orthonormal basis A basis of unit vectors that are perpendicular to one another. In one, the coordinates of a vector are just its dot products with the basis vectors. See Chapter 9.

Osculating circle The circle that fits a curve best at a point, on the inside of the bend, with radius \(1/\kappa\), the radius of curvature. See Chapter 19.

Osculating plane The plane of the tangent \(T\) and principal normal \(N\) of a space curve: the plane the curve most nearly lies in at that point. See Chapter 19.

P

Pappus' theorem If \(A\), \(B\), \(C\) lie on one line and \(A'\), \(B'\), \(C'\) on another, then the three points where \(AB'\) meets \(A'B\), \(AC'\) meets \(A'C\) and \(BC'\) meets \(B'C\) are collinear. See Chapter 18.

Parabola The set of points equally far from a focus and a directrix: the conic with eccentricity \(e = 1\). In standard position it is \(y = x^2/(4p)\). See Chapter 8.

Parabolic transformation A Möbius transformation with exactly one fixed point; it is conjugate to a translation \(z \mapsto z + 1\). See Chapter 13.

Parallax The apparent shift of a near object against distant ones when seen from two places. Measuring it with a known baseline gives the distance, as for the Moon and Mars. See Chapter 7.

Parallel lines Two lines in a plane that never meet, written \(\ell \parallel m\). See Chapter 3.

Parallel postulate Euclid's fifth postulate: if a transversal makes the interior angles on one side add to less than two right angles, the two lines meet on that side. It is equivalent to Playfair's axiom, and it is independent of the other axioms. See Chapter 15.

Parallel transport Moving a tangent vector along a curve in a curved space so that its covariant derivative is zero, keeping it "as constant as possible". Around a closed loop on a curved surface it comes back turned, by an angle equal to the curvature inside. See Chapter 20.

Parallelepiped A slanted box built from three vectors from one corner. Its volume is the absolute value of the scalar triple product of the three vectors. See Chapter 9.

Parallelogram A quadrilateral with both pairs of opposite sides parallel. Its opposite sides and opposite angles are equal and its diagonals bisect each other. See Chapter 3.

Parallelogram law In any parallelogram, the sum of the squares of the four sides equals the sum of the squares of the two diagonals. See Chapter 8.

Parallelogram rule The sum \(\mathbf{u} + \mathbf{v}\) of two vectors drawn from one point is the diagonal of the parallelogram they span. See Chapter 9.

Parametric equations Equations that give the coordinates of a point of a line or curve in terms of a parameter \(t\), such as \(x = x_0 + ta\), \(y = y_0 + tb\), \(z = z_0 + tc\) for a line. See Chapter 9.

Parametrised curve A smooth function \(\gamma\) from an interval to \(\mathbb{R}^n\), thought of as the position of a moving particle at time \(t\). See Chapter 19.

Parametrised surface A smooth map \(X(u, v)\) from a region of the plane into space, with \(X_u\) and \(X_v\) independent: a chart of a surface. See Chapter 19.

Pascal's theorem If a hexagon is inscribed in a conic, the three points where its pairs of opposite sides meet are collinear, on the Pascal line. See Chapter 18.

Pasch's axiom Hilbert's axiom II.4: if a line enters a triangle through one side, not through a vertex, it leaves through one of the other two sides. Euclid used it without stating it. See Chapter 15.

Pentagram The five-pointed star made by joining every second vertex of a regular pentagon. Its turning number is 2 and its point angles add to \(180°\). See Chapter 3.

Perihelion The point of a planet's orbit closest to the Sun; the farthest point is the aphelion. They lie at distances \(a(1 - e)\) and \(a(1 + e)\) from the Sun. See Chapter 8.

Perpendicular Two lines are perpendicular if they meet at a right angle, written \(\ell \perp m\). See Chapter 1.

Perpendicular bisector The line through the midpoint of a segment at right angles to it. It is the locus of points equally far from the two ends. See Chapter 2.

Pi The ratio \(\pi \approx 3.14159\) of the circumference of any circle to its diameter. It is irrational (Lambert, 1760s) and transcendental (Lindemann, 1882). See Chapter 6.

Planar graph A set of points joined by curves that do not cross, drawn in the plane. Flattening a convex polyhedron (a Schlegel diagram) gives one, which is how Euler's formula is proved. See Chapter 11.

Plane A flat surface with no thickness that goes on forever in every direction. Through any three points not on one line there is exactly one plane. See Chapter 1.

Platonic solid A convex polyhedron whose faces are congruent regular polygons, with the same number meeting at every vertex. There are exactly five: tetrahedron, cube, octahedron, dodecahedron and icosahedron. See Chapter 11.

Playfair's axiom Through a point not on a line there is at most one line parallel to it. It is the usual modern form of Euclid's parallel postulate, and equivalent to it. See Chapter 3.

Poincaré conjecture Poincaré's question (1904), proved by Perelman in 2002-2003: every closed, simply connected 3-manifold is a 3-sphere. See Chapter 21.

Poincaré disc The model of the hyperbolic plane inside the unit disc, with lines the diameters and the circular arcs meeting the boundary at right angles, true angles, and lengths \(ds = \frac{2|dz|}{1 - |z|^2}\). See Chapter 17.

Poincaré-Hopf theorem The zeros of a vector field on a closed surface, counted with their indices, add up to the Euler characteristic. The hairy ball theorem is the case of the sphere, where \(\chi = 2 \ne 0\). See Chapter 20.

Point The undefined term that marks a position. It has no length, width or thickness. See Chapter 1.

Point at infinity An extra point \(\infty\) added to the plane so that inversion is defined everywhere: the centre goes to \(\infty\) and \(\infty\) to the centre. In projective geometry there is one point at infinity for each direction of parallel lines. See Chapter 13.

Point group The group of rotation parts (the matrices \(A\)) of the symmetries of a wallpaper pattern or crystal. By the crystallographic restriction it is one of ten finite groups in the plane. See Chapter 10.

Polar coordinates The pair \((r, \theta)\) that locates a point by its distance \(r\) from the origin and the angle \(\theta\) from the positive \(x\)-axis; \(x = r\cos\theta\) and \(y = r\sin\theta\). See Chapter 8.

Polar triangle For a spherical triangle \(ABC\), the triangle formed by the poles of its three sides. Its sides are \(\pi - A\), \(\pi - B\), \(\pi - C\), which turns facts about sides into facts about angles. See Chapter 16.

Polygon A closed chain of segments (its sides) that do not cross one another, with no two neighbouring sides on one line. With \(n\) sides it is an \(n\)-gon. See Chapter 3.

Polyhedron A solid bounded by flat polygons, its faces. Faces meet along edges, and edges meet at vertices. See Chapter 11.

Poncelet-Steiner theorem Everything constructible with straightedge and compass can be constructed with the straightedge alone, given one circle drawn with its centre marked. See Chapter 14.

Porism A statement that a construction either never works or works for every starting position. Steiner's porism about chains of circles is the classic example. See Chapter 13.

Position vector The vector \(\overrightarrow{OP}\) from a chosen origin \(O\) to a point \(P\); its components are the coordinates of \(P\). See Chapter 9.

Postulate A starting fact accepted without proof, an axiom. Euclid used "postulate" for the five facts specific to geometry. See Chapter 1.

Power of a point For a point \(P\) at distance \(d\) from the centre of a circle of radius \(r\), the number \(d^2 - r^2\). Every line through \(P\) meeting the circle at \(A\) and \(B\) has \(PA \cdot PB\) equal to its absolute value. See Chapter 6.

Primitive Pythagorean triple A Pythagorean triple whose three numbers have no common factor, such as \((3, 4, 5)\) or \((5, 12, 13)\). See Chapter 4.

Principal curvatures The largest and smallest normal curvatures \(k_1\) and \(k_2\) at a point of a surface. They occur in two perpendicular principal directions. See Chapter 19.

Principal normal The unit vector \(N = T'/\kappa\) of a curve, perpendicular to the tangent and pointing to the inside of the bend. See Chapter 19.

Prism A polyhedron with two congruent parallel polygons (the bases) joined by parallelograms. If the side faces are rectangles it is a right prism. Its volume is base area times height. See Chapter 11.

Projection (of a vector) The vector projection of \(\mathbf{v}\) onto \(\mathbf{u}\) is the part of \(\mathbf{v}\) along \(\mathbf{u}\), \(\frac{\mathbf{u}\cdot\mathbf{v}}{\mathbf{u}\cdot\mathbf{u}}\mathbf{u}\); its signed length \(\mathbf{u}\cdot\mathbf{v}/|\mathbf{u}| = |\mathbf{v}|\cos\theta\) is the scalar projection. See Chapter 9.

Projective transformation A map of the projective plane, given by an invertible \(3 \times 3\) matrix acting on homogeneous coordinates, that sends lines to lines. Any four points in general position can be sent to any other four. See Chapter 18.

Projective variety The set of common zeros of homogeneous polynomials in projective space: an affine variety completed by its points at infinity. See Chapter 21.

Proper time The time measured by a moving clock's own reading, \(\sqrt{-ds^2}/c\) along its path in spacetime. Freely falling bodies follow paths of greatest proper time. See Chapter 20.

Protractor postulate The starting facts about measuring angles: each angle has a measure between \(0°\) and \(180°\), the measures of adjacent angles add, and a straight angle measures \(180°\). See Chapter 1.

Ptolemy's theorem In a cyclic quadrilateral \(ABCD\), the product of the diagonals equals the sum of the products of opposite sides: \(AC \cdot BD = AB \cdot CD + AD \cdot BC\). Ptolemy used it to build his table of chords. See Chapter 12.

Pyramid A polyhedron with one polygon base and triangles joining its edges to one point, the apex. Its volume is \(\tfrac13\) base area times height. See Chapter 11.

Pythagoras' theorem In a right triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^2 + b^2 = c^2\). Conversely, a triangle whose sides satisfy this has a right angle opposite \(c\). See Chapter 4.

Pythagorean triple Three whole numbers \((a, b, c)\) with \(a^2 + b^2 = c^2\), such as \((3, 4, 5)\). They correspond to the rational points on the unit circle. See Chapter 4.

Q

Quadratrix A curve drawn by two uniform motions, invented by Hippias. It trisects any angle and squares the circle, but it cannot be drawn with straightedge and compass. See Chapter 14.

Quadrilateral A polygon with four vertices joined in order by four sides that do not cross one another. See Chapter 2.

R

Radian The measure of a central angle as the length of the arc it cuts off divided by the radius. A full turn is \(2\pi\) radians, so \(1\) radian \(\approx 57.3°\). See Chapter 7.

Radical axis The line of points that have equal power with respect to two circles. For two crossing circles it is the line through their two common points. See Chapter 6.

Radius The fixed distance from the centre of a circle to its points, or a segment from the centre to a point of the circle. See Chapter 6.

Radius of curvature The radius \(1/\kappa\) of the osculating circle of a curve at a point. See Chapter 19.

Rapidity In the hyperboloid model of relativity, the hyperbolic distance \(d\) between two velocities. The relative speed is \(\tanh d\), and rapidities of motions in a line simply add. See Chapter 17.

Ray The part of a line that starts at a point \(A\), passes through \(B\) and goes on forever, written \(\overrightarrow{AB}\). See Chapter 1.

Real projective plane The set \(\mathbb{RP}^2\) of lines through the origin in \(\mathbb{R}^3\), a line of it being a plane through the origin. It is the ordinary plane with a line at infinity added, and any two of its lines meet in exactly one point. See Chapter 18.

Rectangle A quadrilateral with four right angles. It is a parallelogram with equal diagonals. See Chapter 2.

Reflection The isometry that sends each point to its mirror image in a line \(\ell\) (the mirror): the point on the other side at the same distance, with \(\ell\) the perpendicular bisector of the segment joining them. See Chapter 10.

Reflex angle An angle between \(180°\) and \(360°\), "the outside" of an ordinary angle. See Chapter 1.

Regular curve A parametrised curve whose velocity \(\gamma'(t)\) is never zero. See Chapter 19.

Regular polygon A polygon whose sides are all equal and whose angles are all equal. The angle of a regular \(n\)-gon is \((n - 2) \cdot 180°/n\). See Chapter 3.

Regular tiling A tiling of the plane by copies of one regular polygon meeting edge to edge. In the Euclidean plane there are only three, by triangles, squares and hexagons; in the hyperbolic plane there are infinitely many, \(\{p, q\}\) with \(1/p + 1/q < 1/2\). See Chapter 3.

Relative consistency A theory is consistent if another one is, because a model of the first can be built inside the second. The Klein disc shows that hyperbolic geometry is consistent if Euclidean geometry is. See Chapter 15.

Reparametrisation Changing the parameter of a curve by \(t = \phi(s)\) with \(\phi\) smooth and \(\phi' > 0\): the same path traced at a different speed. Geometric quantities like arc length and curvature do not change. See Chapter 19.

Rhombic dodecahedron The Catalan solid dual to the cuboctahedron, with 12 rhombus faces, 24 edges and 14 vertices. Copies of it fill space, and garnet crystals take its shape. See Chapter 11.

Rhombus A quadrilateral with four equal sides. Its diagonals are perpendicular and bisect each other. See Chapter 2.

Rhumb line A path on the sphere that keeps a constant compass bearing, crossing every meridian at the same angle; also called a loxodrome. It is easy to sail but not a great circle, and on Mercator's map it is straight. See Chapter 16.

Ricci curvature The average of the sectional curvatures of all planes containing a direction \(v\). It measures how the volume of a small cone of geodesics differs from the flat one, and it is the curvature in Einstein's equations. See Chapter 20.

Ricci flow Hamilton's evolution of a metric by \(\partial g/\partial t = -2\,\mathrm{Ric}\), which smooths out curvature as heat smooths out temperature. Perelman used it, with surgery on its singularities, to prove the Poincaré and geometrization conjectures. See Chapter 21.

Riemann curvature tensor The object \(R(X, Y)Z\) built from the connection that measures how much parallel transport around a small loop fails to return a vector to itself. It determines all the sectional, Ricci and scalar curvatures. See Chapter 20.

Riemann sphere The extended complex plane \(\mathbb{C} \cup \{\infty\}\) seen as a sphere through stereographic projection, with \(\infty\) at the north pole. Möbius transformations act on it as conformal maps. See Chapter 13.

Riemann surface A 2-manifold with charts whose transition maps are complex-analytic. Every orientable surface with a Riemannian metric can be made into one. See Chapter 20.

Riemannian manifold A smooth manifold together with a Riemannian metric. See Chapter 20.

Riemannian metric A choice, at each point of a smooth manifold, of an inner product \(g\) on the tangent space, varying smoothly. It gives lengths of curves, angles, volumes and distance. See Chapter 20.

Right angle An angle of exactly \(90°\), a quarter turn, marked with a small square. See Chapter 1.

Right triangle A triangle with one right angle. The side opposite it is the hypotenuse and the other two are the legs. See Chapter 2.

Right-hand rule The rule that fixes the direction of \(\mathbf{u} \times \mathbf{v}\): curl the fingers of the right hand from \(\mathbf{u}\) to \(\mathbf{v}\) and the thumb points along the cross product. See Chapter 9.

Rotation The isometry that turns every point about a fixed centre through a fixed angle. In coordinates about the origin, \((x, y) \mapsto (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)\). See Chapter 10.

S

Saccheri quadrilateral A quadrilateral \(ABCD\) with right angles at \(A\) and \(B\) and equal sides \(AD = BC\). Its summit angles at \(C\) and \(D\) are equal, and they are right, obtuse or acute according to which geometry holds. See Chapter 3.

Saccheri-Legendre theorem In neutral geometry, the angle sum of every triangle is at most \(180°\). See Chapter 15.

SAS Two triangles are congruent if two sides and the angle between them match. Following Hilbert, this book takes it as an axiom, and proves the other tests from it. See Chapter 2.

Scalar A number used to scale a vector: \(t\mathbf{v}\) is \(t\) times as long as \(\mathbf{v}\), reversed if \(t < 0\). See Chapter 9.

Scalar curvature The single function \(S\) got by tracing the Ricci curvature, the sum of \(\mathrm{Ric}(e_i, e_i)\) over an orthonormal basis. It measures how the volume of a small ball differs from the Euclidean one. See Chapter 20.

Scalar triple product The number \(\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})\), the determinant of the three vectors. Its absolute value is the volume of the parallelepiped they span. See Chapter 9.

Scalene triangle A triangle whose three sides all have different lengths. See Chapter 2.

Schläfli symbol The symbol \(\{p, q\}\) for a regular polyhedron or tiling whose faces are \(p\)-gons with \(q\) at each vertex: the cube is \(\{4, 3\}\). See Chapter 11.

Scissors-congruent Two polygons are scissors-congruent (equidecomposable) if one can be cut into finitely many polygons that reassemble into the other. See Chapter 4.

Screw motion An isometry of space that rotates about an axis and slides along it. See Chapter 10.

Secant A line meeting a circle in two points. See Chapter 6.

Second fundamental form The quadratic form \(II = L\,du^2 + 2M\,du\,dv + N\,dv^2\), with \(L = X_{uu}\cdot n\) and so on, that measures how a surface bends away from its tangent plane. The normal curvature in a direction is \(II/I\). See Chapter 19.

Section formula The point dividing segment \(AB\) in the ratio \(m : n\) has position vector \(\frac{n\mathbf{a} + m\mathbf{b}}{m + n}\). See Chapter 9.

Sectional curvature For a 2-dimensional plane \(\sigma\) of tangent vectors at a point, the Gaussian curvature of the surface swept out by geodesics in the directions of \(\sigma\). On a surface it is Gauss's \(K\). See Chapter 20.

Sector The region between two radii of a circle and the arc between them, a slice of pizza. A sector with angle \(\theta\) radians has area \(\tfrac12 r^2\theta\). See Chapter 6.

Segment (of a circle) The region between a chord and the arc it cuts off. See Chapter 6.

Segment (of a line) The part \(AB\) of a line between two points \(A\) and \(B\), including the endpoints. It has a length, written \(AB\). See Chapter 1.

Semi-major axis Half the longest diameter of an ellipse, \(a\); half the shortest is the semi-minor axis \(b\). For an ellipse with eccentricity \(e\), \(b = a\sqrt{1 - e^2}\). See Chapter 8.

Semi-perimeter Half the perimeter of a triangle, \(s = \tfrac12(a + b + c)\). See Chapter 4.

Semi-regular tiling A tiling by regular polygons of more than one kind, edge to edge, with the same polygons in the same cyclic order at every vertex; also called Archimedean. There are eight. See Chapter 3.

Shape operator The linear map of the tangent plane of a surface that sends a direction to the rate of change of the unit normal. Its eigenvalues are the principal curvatures and its determinant is the Gaussian curvature. See Chapter 19.

Shear Sliding the top of a figure along a line parallel to its base. It changes the shape and keeps the area. See Chapter 4.

Shoelace formula The area of a polygon with vertices \((x_1, y_1), \ldots, (x_n, y_n)\) in order is \(\tfrac12\left|\sum (x_i y_{i+1} - x_{i+1} y_i)\right|\). See Chapter 4.

Signed ratio For points \(X\), \(Y\), \(P\) on one line, the ratio \(XP/PY\) of lengths, taken positive if \(P\) is between \(X\) and \(Y\) and negative if it is outside. Ceva's and Menelaus' theorems use it. See Chapter 12.

Similar Two figures are similar if one can be turned into the other by a dilation followed by an isometry: angles are equal and lengths are in a fixed ratio. For triangles, equal angles alone or proportional sides alone are enough. See Chapter 5.

Similarity A map that multiplies all distances by the same ratio \(k > 0\). Every similarity is a dilation followed by an isometry. See Chapter 10.

Similarity tests Two triangles are similar if two angles match (AA), if two sides are in proportion with the angle between them equal (SAS), or if all three sides are in proportion (SSS). See Chapter 5.

Simply connected A space is simply connected if every loop in it can be shrunk to a point within the space. A sphere is; a torus is not. See Chapter 21.

Simson line For a point \(P\) on the circumcircle of a triangle, the feet of the perpendiculars from \(P\) to the three side lines are collinear, on its Simson line. For points off the circle they are not collinear. See Chapter 12.

Sine In a right triangle, the opposite side divided by the hypotenuse; in general, the \(y\)-coordinate of the point of the unit circle at angle \(\theta\). See Chapter 7.

Skew lines Two lines in space that are neither parallel nor meeting, like a road and a bridge that crosses over it. See Chapter 1.

Slope The steepness \(m = \frac{y_2 - y_1}{x_2 - x_1}\) of the line through two points. Parallel lines have equal slopes, and perpendicular lines have slopes whose product is \(-1\). See Chapter 8.

Small circle A circle cut from a sphere by a plane that misses the centre, like a circle of latitude other than the equator. See Chapter 16.

Smooth manifold A topological manifold with an atlas whose transition maps are all smooth (infinitely differentiable). On it, calculus makes sense. See Chapter 20.

Spacetime The 4-dimensional manifold of events of relativity, with a Lorentzian metric. In general relativity matter curves it, and freely falling bodies follow its geodesics. See Chapter 20.

Spherical excess The amount \(E = \alpha + \beta + \gamma - \pi\) by which a spherical triangle's angle sum exceeds \(\pi\). By Girard's theorem the area is \(ER^2\). See Chapter 16.

Spherical triangle Three points on a sphere joined by the shorter great-circle arcs. Its angles are the angles between the planes of the arcs, and its angle sum is always more than \(180°\). See Chapter 16.

Spiral of Archimedes The curve \(r = a\theta\) in polar coordinates, whose turns are equally spaced like the groove of a record. See Chapter 8.

Spiral similarity A similarity that turns about a centre and scales from it. In complex numbers it is \(z \mapsto az + b\) with \(a \ne 1\), and every direct similarity that is not a translation is one. See Chapter 10.

Square A quadrilateral with four equal sides and four right angles: both a rectangle and a rhombus. See Chapter 2.

Square-cube law When a solid is scaled by \(k\), areas grow by \(k^2\) and volumes by \(k^3\). That is why giants would break their bones and small animals lose heat fast. See Chapter 5.

Squaring the circle The Greek problem of constructing a square with the same area as a given circle. It is impossible with straightedge and compass because \(\pi\) is transcendental. See Chapter 14.

SSA The pattern side-side-angle, with the angle not between the two sides. It does not guarantee congruence: two different triangles can fit it. See Chapter 2.

SSS Two triangles are congruent if their three sides match. See Chapter 2.

Steiner's porism If a chain of circles, each touching the next and all touching two given non-crossing circles, closes up for one starting circle, it closes up for every starting circle. See Chapter 13.

Stereographic projection The map from a sphere (minus its north pole \(N\)) to a plane that sends each point \(P\) to where the line \(NP\) meets the plane. It is conformal and sends circles on the sphere to generalised circles. See Chapter 13.

Stewart's theorem If a cevian of length \(d\) from \(A\) cuts side \(BC\) (of length \(a\)) into pieces \(m\) and \(n\), then \(b^2 m + c^2 n = a(d^2 + mn)\). It gives the lengths of medians and angle bisectors. See Chapter 12.

Straight angle An angle of exactly \(180°\), whose arms are opposite rays forming a line. See Chapter 1.

Straightedge The construction tool that draws the line through two points already constructed. It has no marks, so it cannot measure. See Chapter 2.

Subtend An angle subtends (stands on) the arc of a circle that lies between its arms. See Chapter 6.

Supplementary angles Two angles that add up to \(180°\). See Chapter 1.

Symmetry A symmetry of a figure \(X\) is an isometry \(f\) with \(f(X) = X\): after the motion the figure occupies exactly the same place. See Chapter 10.

Symmetry group The group of all symmetries of a figure under composition. A square's is the dihedral group \(D_4\) of 8 symmetries. See Chapter 10.

T

Tangent bundle The disjoint union \(TM\) of all the tangent spaces of a manifold, itself a manifold of dimension \(2n\). A vector field is a smooth section of it. See Chapter 20.

Tangent plane The plane \(T_pS\) spanned by the coordinate vectors \(X_u\) and \(X_v\) of a surface at a point: the plane that touches the surface there. See Chapter 19.

Tangent space The set \(T_pM\) of tangent vectors at a point \(p\) of a manifold, a vector space of the same dimension as \(M\). See Chapter 20.

Tangent (to a circle) A line that meets a circle in exactly one point, the point of contact. It is perpendicular to the radius there, and the two tangents from an outside point have equal length. See Chapter 6.

Tangent (trigonometric) In a right triangle, the opposite side divided by the adjacent side; in general, \(\tan\theta = \sin\theta/\cos\theta\). See Chapter 7.

Tangent vector At a point \(p\) of a manifold, an equivalence class of curves through \(p\) that have the same velocity in some chart. It is the abstract version of an arrow tangent to a surface. See Chapter 20.

Tetrahedron A pyramid with a triangular base: four triangular faces. With edges \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) from one corner its volume is \(\tfrac16|\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})|\). See Chapter 9.

Thales' theorem An angle inscribed in a semicircle is a right angle. It is the special case of the inscribed angle theorem where the central angle is a straight angle. See Chapter 6.

Theorema Egregium Gauss's "remarkable theorem" (1827): the Gaussian curvature of a surface depends only on its first fundamental form, so it is unchanged by bending without stretching. That is why no flat map of the Earth is perfect. See Chapter 19.

Tiling A covering of the whole plane by copies of shapes, with no gaps and no overlaps; also called a tessellation. See Chapter 3.

Tissot indicatrix The small ellipse into which a map sends a tiny circle on the sphere. It shows at a glance how a map distorts shape and area at each place. See Chapter 16.

Topological manifold A space in which every point has a neighbourhood that is homeomorphic to an open set of \(\mathbb{R}^n\); \(n\) is its dimension. See Chapter 20.

Torsion The rate \(\tau\) at which a space curve twists out of its osculating plane, from \(B' = -\tau N\). A plane curve has zero torsion, and curvature and torsion together determine a curve up to a rigid motion. See Chapter 19.

Tower law For fields \(F \subseteq K \subseteq L\), the degrees multiply: \([L : F] = [L : K][K : F]\). It shows that every constructible number has degree a power of 2. See Chapter 14.

Transcendental number A number that is not a root of any nonzero polynomial with rational (or integer) coefficients. Lindemann proved in 1882 that \(\pi\) is transcendental. See Chapter 14.

Transformation A rule \(f\) that sends every point of the plane to a point \(f(P)\), its image, so that every point is the image of exactly one point. See Chapter 10.

Transition map Where two charts of a manifold overlap, the map \(\psi \circ \varphi^{-1}\) that changes one set of coordinates into the other. See Chapter 20.

Translation The isometry \(P \mapsto P + \mathbf{v}\) that slides every point by the same vector \(\mathbf{v}\). See Chapter 10.

Transversal A line that crosses two other lines. At the two crossings it makes eight angles. See Chapter 3.

Trapezoid A quadrilateral with at least one pair of parallel sides (British: trapezium). Its area is half the sum of the parallel sides times the distance between them. See Chapter 2.

Triangle Three points not on one line, \(A\), \(B\), \(C\), with the three segments joining them (its sides). The side opposite \(A\) is written \(a = BC\), and so on. See Chapter 2.

Triangle inequality In any triangle, each side is shorter than the sum of the other two: \(AC < AB + BC\). A straight path is the shortest. See Chapter 2.

Triangulation Mapping a country by measuring one baseline very carefully and then only angles: each new point is fixed by the angles from two known points, using the law of sines. See Chapter 7.

Trisecting the angle The Greek problem of dividing any angle into three equal parts with straightedge and compass. It is impossible in general: \(60°\) cannot be trisected, because \(\cos 20°\) has degree 3. See Chapter 14.

Truncated icosahedron The Archimedean solid made by cutting each corner of an icosahedron a third of the way along its edges, with 12 pentagons and 20 hexagons: the shape of a football and of the carbon-60 molecule. See Chapter 11.

Turning number The number of times the direction of a closed path turns round as you go round it once. For a simple closed curve it is \(\pm 1\), so the total signed curvature \(\int\kappa\,ds\) is \(\pm 2\pi\). See Chapter 19.

Two-column proof A school layout of a proof with each statement written next to its reason. See Chapter 1.

U

Ultraparallels In the hyperbolic plane, the lines through a point that neither meet a given line nor approach it at an end. Each has a common perpendicular with the given line, and they move apart on both sides of it. See Chapter 17.

Undefined terms The few basic words, in the plane point and line and the relations lies on and between, that a modern geometry leaves undefined; the axioms say how they behave. See Chapter 1.

Uniformization theorem Every closed surface carries a metric of constant curvature \(+1\), \(0\) or \(-1\): the sphere is spherical, the torus flat, and every surface with two or more holes hyperbolic. See Chapter 20.

Unit circle The circle with centre the origin and radius 1. The point reached by turning through \(\theta\) is \((\cos\theta, \sin\theta)\), which defines sine and cosine for all angles. See Chapter 7.

Unit vector A vector of length 1. Any nonzero \(\mathbf{v}\) gives one, \(\mathbf{v}/|\mathbf{v}|\), its direction. See Chapter 9.

Upper half-plane The model of the hyperbolic plane on \(\{z = x + iy : y > 0\}\), with lines the vertical half-lines and the semicircles centred on the real axis, true angles, and lengths \(ds = |dz|/y\). See Chapter 17.

V

Vanishing point The point of a perspective picture where the images of a family of parallel lines meet: the image of their common point at infinity. See Chapter 18.

Vector A displacement: a size together with a direction, drawn as an arrow. Two arrows with the same length and direction are the same vector, wherever they are drawn. See Chapter 9.

Vector equation of a line The equation \(\mathbf{r}(t) = \mathbf{p} + t\mathbf{d}\) of the line through the point with position vector \(\mathbf{p}\) in the direction \(\mathbf{d}\); \(t\) is the parameter. See Chapter 9.

Vector field A smooth choice of a tangent vector at every point of a manifold, like the wind at every point of the Earth. See Chapter 20.

Vector space A set of objects that can be added and scaled, obeying the eight rules that arrows obey. It is the starting point of linear algebra. See Chapter 9.

Vertex The shared endpoint of the two arms of an angle; also a corner of a polygon or a polyhedron. See Chapter 1.

Vertex configuration The list of polygons around a vertex of a polyhedron or tiling, in order: the cube is \(4.4.4\) and the cuboctahedron \(3.4.3.4\). See Chapter 11.

Vertical angles The two opposite angles formed where two lines cross ("vertical" means "at the vertex"). They are always equal. See Chapter 1.

Voronoi diagram The tiling of the plane by the Voronoi cells of a set of sites: the cell of a site is the set of points closer to it than to any other site. See Chapter 21.

W

Wallpaper group The symmetry group of a wallpaper pattern, one that repeats in two independent directions. There are exactly seventeen. See Chapter 10.

Z

Zero vector The vector \(\mathbf{0}\), the movement "stay put". It is the only vector with no direction. See Chapter 9.