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Slide 01
Geometry.
- Vol. XIII · Deck 02 · The Deck Catalog
- From Euclid's Elements to the einstein tile of 2023. Twenty-three centuries on the shapes of space — Euclidean, hyperbolic, projective, differential, algebraic.
- Foundingc. 300 BCE
- Postulatesfive
- Pages32
Slide 02
OpeningWhat geometry is.
- LedeII
- Geometry is the mathematics of shape, space, and the relations between them. It is the oldest branch of the discipline that still has a recognisable name.
- The Egyptians measured fields after the Nile flooded; the Babylonians solved tablet problems on circle areas; the Greeks proved theorems. The Greek invention of proof — the demonstration that a fact follows necessarily from agreed premises — is the founding act of the subject and arguably of mathematics itself.
- Twenty-three centuries on, geometry has split into a dozen specialised disciplines. This deck traces the line from Pythagoras to Penrose and the einstein tile.
- Vol. XIII— ii —
Slide 03
Chapter IEgypt and Babylon.
- OriginsIII
- The Egyptian Rhind papyrus (c. 1650 BCE), copied by the scribe Ahmes from a still earlier source, contains a method for the area of a circle equivalent to taking π ≈ 256/81 ≈ 3.16. Plimpton 322, a Babylonian tablet from c. 1800 BCE, lists fifteen Pythagorean triples in cuneiform — a thousand years before Pythagoras was born.
- This pre-Greek geometry was practical: surveying after the Nile floods, building pyramids and ziggurats, computing volumes of grain silos. The recipes were correct often enough; why they were correct was not the question being asked.
- Herodotus credits the Egyptians with originating geometry; the etymology supports him — geo-metria is "earth-measurement." But the proof tradition was something the Greeks invented from scratch.
- Geometry · Origins— iii —
Slide 04
Chapter IIThales of Miletus.
- ThalesIV
- Thales (c. 624 – c. 546 BCE) is the first Greek mathematician we have a name for. Tradition credits him with five theorems: a circle is bisected by its diameter; the base angles of an isosceles triangle are equal; vertical angles are equal; two triangles with one side and two adjacent angles equal are congruent; an angle inscribed in a semicircle is a right angle (this last is "Thales's theorem").
- The historical detail is uncertain — most of what we have about Thales comes from Aristotle, Eudemus, and Proclus, writing centuries later. What's important is that with Thales, mathematics began to look like proof rather than recipe.
- Thales reportedly measured the height of an Egyptian pyramid by comparing its shadow to the shadow of a stick of known height — the first recorded use of similar triangles for indirect measurement.
- Geometry · Thales— iv —
Slide 05
Chapter IIIThe Pythagorean theorem.
- PythagorasV
- For a right triangle with legs a, b and hypotenuse c: a² + b² = c². The result was empirically known to the Babylonians and Indians a millennium earlier; the Greeks proved it.
- Pythagoras of Samos (c. 570 – c. 495 BCE) and his school combined mathematics, music, and mysticism. They discovered the connection between numerical ratios and musical consonance, and the existence of incommensurable magnitudes — the diagonal of a unit square (length √2) cannot be expressed as a ratio of integers.
- Tradition holds that the Pythagorean Hippasus was drowned at sea for revealing irrationality. The story is almost certainly apocryphal but captures something true: the existence of irrationals was a crisis. The Greek geometric framework had to be rebuilt — by Eudoxus's theory of proportions — to accommodate them.
- Geometry · Pythagoras— v —
Slide 06
Chapter IVThe Elements.
- EuclidVI
- Euclid's Elements (c. 300 BCE), written at Alexandria under the early Ptolemies, is the most successful textbook in history. Thirteen books, 465 propositions. After the Bible, it is the most-printed book in the Western canon.
- Books I–IV cover plane geometry; V–VI proportions; VII–IX number theory (including the proof that there are infinitely many primes); X irrationals; XI–XIII solid geometry, ending with the construction and classification of the five regular polyhedra (the Platonic solids).
- The astonishing achievement is structural. Every proposition is derived from earlier propositions, all the way back to five postulates and a small number of common notions. For two thousand years Elements was the model of what rigorous knowledge should look like — Spinoza's Ethics, Newton's Principia, and Hilbert's Grundlagen der Geometrie all imitate its form.
- Geometry · Euclid— vi —
Slide 07
Chapter VThe five postulates.
- PostulatesVII
- Euclid's foundational assumptions, in modern paraphrase:
- I. A straight line can be drawn from any point to any other.
- II. A finite straight line can be extended continuously.
- III. A circle can be drawn with any centre and radius.
- IV. All right angles are equal to one another.
- V. If a line meets two others such that the interior angles on one side sum to less than two right angles, the two lines, extended sufficiently, meet on that side. (The "parallel postulate.")
- The fifth was suspect from the start — it sounds like a theorem rather than a primitive assumption. For two millennia mathematicians tried to derive it from the others. Every attempt failed. The eventual answer (see Chapter XV) was that the parallel postulate is genuinely independent: alternative geometries exist where it fails.
- Geometry · Postulates— vii —
Slide 08
Chapter VICompass and straightedge.
- ConstructionsVIII
- The Greek geometric programme restricted construction tools to two: an unmarked straightedge and a collapsing compass. With these instruments and finitely many steps, what shapes can you build?
- Euclid's Elements proves you can bisect a line segment, bisect an angle, drop a perpendicular, construct a regular triangle, square, pentagon, and hexagon, and combine these to construct regular polygons of 15 sides and various powers-of-2 multiples.
- Gauss (1796, age 19) found the next step in two thousand years: a regular 17-gon is constructible. He proved more generally that a regular n-gon is constructible exactly when n is a power of 2 times a product of distinct Fermat primes (primes of the form 2^(2^k) + 1). The result is so beautiful Gauss requested a 17-gon on his tombstone — the stonemason refused, judging it indistinguishable from a circle.
- Geometry · Constructions— viii —
Slide 09
Chapter VIIThree classical problems.
- Three impossiblesIX
- Three constructions the Greeks attempted and failed at: squaring the circle (build a square with the area of a given circle), doubling the cube (build a cube with twice the volume of a given one), trisecting an arbitrary angle.
- The 19th century proved all three impossible with compass and straightedge. Pierre Wantzel (1837) settled cube-doubling and angle-trisection by showing the resulting field extensions have degree 3, while compass-and-straightedge constructions can only produce extensions of degree a power of 2. Ferdinand von Lindemann (1882) settled circle-squaring by proving π is transcendental — not the root of any polynomial with rational coefficients, hence not constructible.
- The negative results were as important as positive ones would have been. They showed mathematics can establish what cannot be done, with the same certainty as what can.
- Geometry · Impossibles— ix —
Slide 10
Chapter VIIIConic sections.
- ApolloniusX
- Apollonius of Perga (c. 240 – c. 190 BCE), the "great geometer," wrote eight books on conic sections — the curves obtained by cutting a cone with a plane. Books I–IV survive in Greek, V–VII in Arabic translation, VIII is lost.
- The classification: circle (cut perpendicular to the axis), ellipse (cut at angle less than the cone's half-angle), parabola (cut parallel to the slant side), hyperbola (cut at angle greater than the half-angle, producing two branches). Apollonius coined the names — from elleipsis (deficiency), parabolē (equal application), hyperbolē (excess) — referring to a Greek geometric procedure for constructing them.
- The conics turned out to be everywhere. Kepler (1609) discovered planetary orbits are ellipses; reflectors of all kinds use parabolic cross-sections; the trajectories of unbound bodies in inverse-square fields are hyperbolas. Apollonius gave physics a head start of nineteen hundred years.
- Geometry · Apollonius— x —
Slide 11
Chapter IXArchimedes of Syracuse.
- ArchimedesXI
- Archimedes (c. 287 – c. 212 BCE), the greatest mathematician of antiquity. He proved the surface area of a sphere is two-thirds that of the circumscribing cylinder; the volume of a sphere is two-thirds that of the cylinder. He requested the sphere-and-cylinder figure on his tombstone (Cicero claims to have found and restored it in 75 BCE).
- His method of exhaustion — bracketing curved regions by polygons of more and more sides — is the direct ancestor of integration. He used it to compute the area of a parabolic segment, the surface area of a sphere, and a value of π between 3 + 10/71 and 3 + 1/7.
- The 1906 rediscovery of the Archimedes Palimpsest revealed his lost work The Method, in which he describes how he discovered the sphere result heuristically — by imagining mechanical balances of infinitesimal slices. He then proved the result rigorously, but the heuristic reasoning was suppressed for two millennia.
- Euclid of Alexandria (c. 300 BCE) — author of the Elements
- Geometry · Archimedes— xi —
Slide 12
Chapter XAnalytic geometry.
- DescartesXII
- René Descartes's La Géométrie (1637) — published as an appendix to his Discourse on Method — fused algebra and geometry. Place a point in the plane by an ordered pair (x, y); describe a curve by an equation; reduce geometric questions to algebraic manipulations.
- The unification was decisive. A circle of radius r centred at the origin is now x² + y² = r²; a line is y = mx + b; a parabola is y = ax² + bx + c. Pierre de Fermat developed the same ideas independently and in some respects earlier, but his work circulated only in manuscript.
- The Cartesian coordinate system is now so deeply embedded in mathematical thought that we forget what an invention it was. Calculus, complex analysis, modern algebraic geometry, and the entire discipline of graphical computing are downstream of one Frenchman's appendix.
- Geometry · Descartes— xii —
Slide 13
Chapter XIProjective geometry.
- ProjectiveXIII
- Renaissance painters needed mathematics to render three-dimensional space on a flat canvas. Filippo Brunelleschi's 1413 demonstration of linear perspective and Leon Battista Alberti's 1435 De Pictura began the formal theory.
- Girard Desargues (1591–1661) generalised perspective into a new geometry in which parallel lines do meet — at a "point at infinity." His Brouillon project (1639) was nearly forgotten until Poncelet revived projective geometry in the 19th century. Desargues's theorem: if two triangles are in perspective from a point, the corresponding sides meet in three collinear points.
- Blaise Pascal, age 16, proved his "mystic hexagram" theorem (1640): the three pairs of opposite sides of a hexagon inscribed in a conic meet in three collinear points. The 19th-century revival under Jean-Victor Poncelet, Felix Klein, and Arthur Cayley made projective geometry the natural framework — and the foundation of computer graphics today, where every viewing transform uses homogeneous coordinates.
- Geometry · Projective— xiii —
Slide 14
Chapter XIIThe parallel postulate falls.
- Non-EuclideanXIV
- For two millennia mathematicians tried to derive Euclid's fifth postulate from the other four. Saccheri (1733) deduced consequences from its negation, hoping to find a contradiction. He did not. Lambert (1766) went further. Both stopped short of accepting what they were finding.
- The decisive step came in the 1820s, made independently by Nikolai Lobachevsky (Kazan) and János Bolyai (Hungary). Both built consistent geometries in which through a point not on a given line, infinitely many parallel lines pass. Carl Friedrich Gauss had reached the same conclusion privately decades earlier but published nothing — "I fear the screams of the Boeotians," he wrote.
- The discovery did not destroy Euclid; it relativised him. Euclidean geometry describes flat space; Bolyai–Lobachevsky geometry describes a different consistent geometry. Both are mathematically valid. Which describes physical space became an empirical question.
- Geometry · Non-Euclidean— xiv —
Slide 15
Chapter XIIIHyperbolic geometry.
- HyperbolicXV
- The Bolyai–Lobachevsky geometry, in which the angles of a triangle sum to less than 180°. The deficit is proportional to the triangle's area — a remarkable rigidity: in hyperbolic space, the size of a triangle is determined by its angles.
- Eugenio Beltrami (1868) gave the first concrete model: the surface of constant negative curvature called the pseudosphere realises hyperbolic geometry locally. Henri Poincaré's disk model and upper-half-plane model (1880s) made the geometry visually accessible — and provide the natural setting for much of modern complex analysis.
- M. C. Escher's Circle Limit woodcuts (1958–1960), made in collaboration with the Canadian geometer H. S. M. Coxeter, are among the most widely-seen visualisations of hyperbolic tilings — repeating patterns that fit infinitely many congruent tiles into a finite disk.
- Geometry · Hyperbolic— xv —
Slide 16
Chapter XIVSpherical geometry.
- SphericalXVI
- The other non-Euclidean geometry — older than Euclid's, in fact, since spherical geometry is what one needs for navigation and astronomy. Menelaus of Alexandria's Sphaerica (c. 100 CE) is the first systematic treatment.
- On the sphere, a "straight line" is a great circle (geodesic). The angles of a triangle sum to more than 180°; the excess is the area times the curvature. There are no parallel lines — every pair of great circles intersects.
- Bernhard Riemann's 1854 Habilitation address generalised these examples: a Riemannian manifold has a curvature, and the curvature determines the local geometry. Spherical geometry is the constant-positive-curvature case, hyperbolic the constant-negative, Euclidean the zero-curvature limit between them. The three classical geometries became three points on a continuum.
- Geometry · Spherical— xvi —
Slide 17
Chapter XVDifferential geometry.
- RiemannXVII
- Bernhard Riemann's 1854 Habilitation at Göttingen, "On the Hypotheses Which Lie at the Foundations of Geometry," reconstructed the discipline. A Riemannian manifold is a space that locally looks like Euclidean space, with a smoothly-varying inner product (the metric tensor) defining lengths and angles. From the metric, every geometric quantity follows: geodesics, curvature, volume.
- The framework subsumed the classical geometries (the metric tensor's signature distinguishes spherical, Euclidean, hyperbolic) and absorbed Gauss's Theorema Egregium (the curvature of a surface depends only on its intrinsic metric, not on how it sits in ambient space).
- Gregorio Ricci-Curbastro and Tullio Levi-Civita developed the tensor calculus (1900) that makes Riemannian geometry computationally tractable. Albert Einstein needed the framework, almost off the shelf, when general relativity was assembled in 1915.
- Geometry · Riemann— xvii —
Slide 18
Chapter XVIKlein's program.
- ErlangenXVIII
- Felix Klein's 1872 inaugural lecture at Erlangen — the Erlanger Programm — proposed that every geometry can be characterised by its group of symmetries.
- Euclidean geometry is the geometry preserved by rigid motions (translations, rotations, reflections); affine geometry is preserved by affine transformations; projective geometry by projective transformations. Move up the hierarchy of groups, lose properties (length, then parallelism, then…), gain a more inclusive geometry.
- The Erlangen view reduced "what is a geometry" to "what is a group acting on a space." It unified projective, affine, hyperbolic, and Euclidean geometry as different group actions on the same underlying set. The view dominates 20th-century geometry; it is also a clean prediction of how physics would later organise itself, with relativistic spacetimes characterised by their Lorentz/Poincaré symmetry groups.
- Geometry · Erlangen— xviii —
Slide 19
Chapter XVIIHilbert's 1900 problems.
- HilbertXIX
- At the 1900 International Congress of Mathematicians in Paris, David Hilbert presented 23 problems for the new century. Several were geometric. The third (Dehn solved it the same year, negatively) asked whether two polyhedra of equal volume can always be cut and reassembled into one another — they cannot. The fifth, on Lie groups as topological groups, was settled by Montgomery, Zippin, and Gleason (1952).
- Hilbert's 1899 Grundlagen der Geometrie reconstructed Euclidean geometry on a genuinely rigorous axiomatic basis — replacing Euclid's intuitive primitives with formal undefined terms (point, line, plane) and 21 axioms grouped under incidence, order, congruence, parallels, and continuity.
- The famous Hilbert remark — "one must be able to say at all times: instead of points, straight lines, and planes, tables, chairs, and beer mugs" — captures the formalist insight: theorems depend on the axioms, not on the meaning of the words.
- Geometry · Hilbert— xix —
Slide 20
Chapter XVIIITopology emerges.
- TopologyXX
- What survives if you abandon distance, angle, and even straightness — keeping only continuity? The answer is topology, the geometry of rubber-sheet deformations.
- The subject crystallised in the late 19th century. Henri Poincaré's Analysis Situs (1895) introduced fundamental groups and homology; Felix Hausdorff's 1914 Grundzüge der Mengenlehre formalised the abstract topology of metric and topological spaces.
- The Euler characteristic V − E + F for polyhedra — Euler 1758, Descartes earlier — turns out to be a topological invariant: it depends only on the surface's genus, not on how the polyhedron is constructed. The classification of closed surfaces (sphere, torus, double torus, …) by genus is the prototype of all subsequent classifications in topology.
- Geometry · Topology— xx —
Slide 21
Chapter XIXCoxeter and 20th-century geometry.
- CoxeterXXI
- By the mid-20th century, geometry had largely been absorbed by topology, algebra, and analysis. Harold Scott MacDonald Coxeter (1907–2003) at Toronto kept the classical tradition alive, working on regular polytopes, reflection groups, and what we now call Coxeter groups — abstract groups generated by reflections.
- Coxeter's books — Regular Polytopes (1947), Introduction to Geometry (1961), Projective Geometry (1964) — taught two generations of mathematicians that synthetic geometry was alive. He collaborated with M. C. Escher on the hyperbolic Circle Limit works.
- The Coxeter groups turned out to be central in unexpected places: as Weyl groups in Lie theory, as buildings in geometric group theory, as the symmetry groups of regular tilings (including the H₃ Coxeter group's role in quasicrystal symmetry). The 20th century rediscovered classical geometry through algebraic eyes.
- The Mandelbrot set — geometry at the boundary of order and chaos
- Geometry · Coxeter— xxi —
Slide 22
Chapter XXPenrose tilings.
- PenroseXXII
- In 1974 Roger Penrose published a tiling of the plane using only two tile shapes (the "kite" and the "dart") that necessarily fits together aperiodically — no translational symmetry, but a five-fold rotational symmetry that classical crystallography had declared impossible.
- Earlier aperiodic tilings — Berger (1966) used 20,426 tiles, then 104; Robinson (1971) used six. Penrose's two-tile result was the spectacular minimum.
- In 1984 Dan Shechtman discovered that aluminium-manganese alloys can grow as quasicrystals — physical solids exhibiting Penrose-like symmetries that crystallography had said were forbidden. The discovery cost him professional reputation initially (Linus Pauling: "There is no such thing as quasicrystals, only quasi-scientists") and won him the 2011 Nobel Prize in Chemistry.
- Geometry · Penrose— xxii —
Slide 23
Chapter XXIAn aperiodic monotile.
- The einsteinXXIII
- Could a single shape — one tile — tile the plane only aperiodically? The question stood open for decades. Penrose's two-tile result was a near miss; reducing to one was widely suspected impossible.
- In March 2023, David Smith (a retired British print technician) identified a 13-sided shape that appeared to do it. Working with Craig Kaplan, Joseph Myers, and Chaim Goodman-Strauss, the team published a proof: the "hat" tile is a true aperiodic monotile — an "einstein," from German ein Stein, "one stone."
- A subsequent May 2023 paper introduced a related family of "spectres," monotiles that are aperiodic without requiring reflected copies. The discovery is one of the most striking elementary mathematical results of the 21st century — a shape simple enough to draw on a napkin, settling a long-open question.
- Geometry · The einstein— xxiii —
Slide 24
Chapter XXIIComputers and geometry.
- Computer-aidedXXIV
- The four-colour theorem (every planar map can be coloured with four colours so adjacent regions differ) was proved in 1976 by Kenneth Appel and Wolfgang Haken at the University of Illinois — the first major theorem with a proof that depended on computer verification of 1,936 cases. Many mathematicians initially refused to accept it as a "real" proof.
- Thomas Hales's 1998 proof of Kepler's conjecture (the densest packing of equal spheres is the face-centred cubic, the orange-stand arrangement) similarly relied on extensive computer computation. He spent the next 17 years on the Flyspeck project, producing a fully formal verification in the HOL Light proof assistant in 2014.
- Computational geometry as a discipline emerged in the 1970s — convex hulls, Voronoi diagrams, Delaunay triangulations. It is now indispensable in graphics, GIS, robotics, and computer-aided design. Geometric questions that were intractable by hand are routine on a laptop.
- Geometry · Computer-aided— xxiv —
Slide 25
Chapter XXIIIGeometry as physics.
- RelativityXXV
- Einstein's general relativity (1915) is a geometric theory. Spacetime is a four-dimensional pseudo-Riemannian manifold; matter and energy curve it; what we call gravity is the resulting curvature. The field equations G_{μν} = 8πG·T_{μν} equate a geometric tensor (Einstein) with a physical one (stress-energy).
- The 1919 solar eclipse expedition led by Arthur Eddington measured starlight bending around the sun and confirmed Einstein's prediction quantitatively. Newspapers ran headlines about geometry being a contingent property of the universe.
- Subsequent confirmations: gravitational time dilation (Pound-Rebka 1959, GPS satellites), gravitational lensing, gravitational waves (LIGO 2015 — direct detection of ripples in spacetime curvature from merging black holes). General relativity remains the best-tested theory of gravity; its geometric core is the central insight.
- Geometry · Relativity— xxv —
Slide 26
Chapter XXIVAlgebraic geometry.
- AlgebraicXXVI
- The geometry of varieties — sets defined by polynomial equations. x² + y² = 1 is the unit circle; y² = x³ + ax + b is an elliptic curve; the zero set of a system of polynomials in many variables is a variety in higher dimension.
- The 19th-century Italian school (Castelnuovo, Enriques, Severi) classified surfaces. The 20th century's Oscar Zariski, André Weil, and decisively Alexander Grothendieck rebuilt the subject on functorial foundations: schemes generalise varieties; sheaves replace pointwise data; cohomology gives the invariants.
- Major modern landmarks: the proof of the Weil conjectures (Deligne 1973–74), the proof of Fermat's last theorem (Wiles 1995, by way of modularity for elliptic curves), the resolution of the Mordell conjecture (Faltings 1983). Algebraic geometry is now the central organising language of much of pure mathematics.
- Geometry · Algebraic— xxvi —
Slide 27
Chapter XXVTropical geometry.
- TropicalXXVII
- A young subject — emerging in the 1990s — that replaces ordinary addition and multiplication with min and addition. The "tropical semiring" has a ⊕ b = min(a, b) and a ⊗ b = a + b.
- Tropical varieties are piecewise-linear analogues of algebraic varieties — a tropical curve is a graph in the plane. Surprisingly, many algebraic-geometric theorems have direct tropical analogues, often easier to prove. The translation works because logarithm-and-limit converts ordinary algebraic geometry into the tropical setting.
- The name "tropical" is a tribute to the Brazilian mathematician Imre Simon, one of the early developers; the geography is honorary. Applications now reach into phylogenetics, optimisation, and statistical mechanics. The subject is one of the few major new branches of geometry to appear in the last fifty years.
- Geometry · Tropical— xxvii —
Slide 28
Chapter XXVISymplectic geometry.
- SymplecticXXVIII
- The geometry of phase space in classical mechanics. The state of a mechanical system with n degrees of freedom is a point in 2n-dimensional phase space (positions and momenta); time evolution preserves a particular geometric structure called the symplectic form.
- The structure was implicit in Hamilton's 1830s reformulation of mechanics; Hermann Weyl coined the term "symplectic" in 1939 (from Greek sym-plektos, "intertwined") to replace the more cumbersome "complex" of complex numbers, which had grown ambiguous.
- The subject was relatively dormant until the 1980s, when Mikhail Gromov's discovery of pseudo-holomorphic curves and the symplectic non-squeezing theorem opened a new field. The connection to mirror symmetry (Kontsevich 1994) connects symplectic geometry to algebraic geometry by way of string theory, in one of the most active research areas in mathematics today.
- Geometry · Symplectic— xxviii —
Slide 29
Chapter XXVIITwenty essentials.
- Reading listXXIX
- c.300 BCEElementsEuclid
- c.225 BCEConicsApollonius
- 1637La GéométrieDescartes
- 1827Disquisitiones GeneralesGauss
- 1854On the Hypotheses…Riemann
- 1872Erlangen ProgramKlein
- 1899Grundlagen der GeometrieHilbert
- 1947Regular PolytopesCoxeter
- 1961Introduction to GeometryCoxeter
- 1963Foundations of Differential GeometryKobayashi & Nomizu
- 1973GravitationMisner, Thorne, Wheeler
- 1976Differential Geometry of Curves and Surfacesdo Carmo
- 1977Algebraic GeometryHartshorne
- 1989The Mathematical ExperienceDavis & Hersh
- 1994Tilings and PatternsGrünbaum & Shephard
- 1995Visual Complex AnalysisNeedham
- 2002The Road to RealityPenrose
- 2008Mathematics and Its HistoryStillwell
- 2014An Imaginary TaleNahin
- 2023An Aperiodic Monotile (paper)Smith, Myers, et al.
- Geometry · Reading list— xxix —
Slide 30
Chapter XXVIIIWatch & read.
- Watch & ReadXXX
- ↑ Euclid explains the point of Geometry · the Elements, gently
- More on YouTube
- Watch · Non-Euclidean Geometry · the Hyperbolica devlog
- Watch · 3Blue1Brown · How to lie using visual proofs
- Geometry · Watch & Read— xxx —
Slide 31
Chapter XXIXWhere to begin.
- ClosingXXXI
- For Euclid: read Heath's translation (1908, still in print) and Hartshorne's Geometry: Euclid and Beyond for a modern axiomatic redo.
- For modern differential geometry: do Carmo's Differential Geometry of Curves and Surfaces is the standard introduction; for general relativity, Misner, Thorne, & Wheeler's Gravitation remains unmatched as a one-thousand-page tour.
- For the contemplative angle: Coxeter's Introduction to Geometry teaches synthetically; Penrose's The Road to Reality embeds geometry in the broader fabric of physics; Stillwell's Mathematics and Its History tracks how the subject evolved.
- The reward is to see what you already see, differently. Every triangle you draw inscribes a small example of a deep theorem. The history of geometry is the history of noticing this — and recognising that the noticing matters.
- Geometry · Closing— xxxi —
Slide 32
The end of the deck.
- ColophonXXXII
- Geometry — Volume XIII, Deck 02 of The Deck Catalog. Set in Cormorant Garamond with Inter for metadata. Cream paper #f4eed8; ink black with gold and sky-blue accents.
- Thirty leaves on the mathematics of shape. From a Greek scribe's Elements to a retired technician's "hat." Geometry remains the most visually direct of the mathematical disciplines — and one of the most surprising.
- FINIS
- ↑ Vol. XIII · Math · Deck 02
