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Geometry — A Drafting Table Deck

A Drafting Table Deck · XIII Plates Geometry / The science of shape From the surveyor’s rope to the curvature of spacetime — thirty-five...

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A Drafting Table Deck · XIII Plates Geometry / The science of shape From the surveyor’s rope to the curvature of spacetime — thirty-five centuries of measuring the world. Key sections include: Geometry /; The science of shape; II. Rope-Stretchers & Star-Watchers; III. Pythagoras of Samos; IV. Euclid’s Elements; V. The Five Postulates; VI. The Parallel Problem; VII. The Crack in the Plane; VIII. Riemann’s Manifolds; IX. Gravity is Geometry.

Key sections

  • 01Geometry /
  • 02The science of shape
  • 03II. Rope-Stretchers & Star-Watchers
  • 04III. Pythagoras of Samos
  • 05IV. Euclid’s Elements
  • 06V. The Five Postulates
  • 07VI. The Parallel Problem
  • 08VII. The Crack in the Plane
  • 09VIII. Riemann’s Manifolds
  • 10IX. Gravity is Geometry
  • 11X. Topology — Shape Without Size
  • 12XI. Fractals — The Roughness of Things
  • 13XII. The Modern Workshop
  • 14XIII. Plates & Pointers

Topics covered

catalogmathgeometry

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Slide outline
  1. 01Geometry /
  2. 02The science of shape
  3. 03II. Rope-Stretchers & Star-Watchers
  4. 04III. Pythagoras of Samos
  5. 05IV. Euclid’s Elements
  6. 06V. The Five Postulates
  7. 07VI. The Parallel Problem
  8. 08VII. The Crack in the Plane
  9. 09VIII. Riemann’s Manifolds
  10. 10IX. Gravity is Geometry
  11. 11X. Topology — Shape Without Size
  12. 12XI. Fractals — The Roughness of Things
  13. 13XII. The Modern Workshop
  14. 14XIII. Plates & Pointers
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Presentation Transcript

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Slide 01

Geometry/

  • A Drafting Table Deck · XIII Plates
  • The science of shape
  • From the surveyor’s rope to the curvature of spacetime — thirty-five centuries of measuring the world.
  • Plate I of XIII
  • Use ← → or click
  • Press F for fullscreen
Slide 02

II.Rope-Stretchers & Star-Watchers

  • Egypt and Babylon, c. 2000 – 600 BC
  • Geometry begins in mud. Each year the Nile flooded and erased the boundaries of the fields; harpedonaptai — rope-stretchers — reset them with knotted cords, recovering rectangles from silt.
  • In Babylon, scribes pressed cuneiform tablets with sexagesimal arithmetic, computing the rising of Venus and the diagonal of a square (YBC 7289 gives √2 to six decimal places).
  • Surveying — restoring property after the flood
  • Astronomy — predicting eclipses, tracking planets
  • Architecture — pyramids aligned to true north within arc-minutes
  • Fig. 1 · The rope-stretcher’s 3-4-5 right angle
Slide 03

III.Pythagoras of Samos

  • c. 570 – 495 BC · the first proof, and a brotherhood that swore by it
  • The Pythagoreans were half mathematicians, half mystics — they believed number was the substance of the cosmos. Their crowning theorem was older than they were (Babylonian tablets knew it), but they gave it something new: a proof.
  • Proposition I.47
  • In any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
  • a2 + b2 = c2
  • Fig. 2 · Squares on the three sides — Euclid I.47
Slide 04

IV.Euclid’s Elements

  • Alexandria, c. 300 BC · the most influential textbook ever written
  • Working in the Library of Alexandria under Ptolemy I, Euclid gathered three centuries of Greek mathematics and rebuilt it from the ground up. The Elements — thirteen books, 465 propositions — was the standard text for over two thousand years.
  • Its method was revolutionary: from a handful of definitions, postulates, and common notions, every theorem follows by deduction alone. Mathematics became, for the first time, an axiomatic science.
  • Books I–VI — plane geometry, ending in similar figures
  • Books VII–X — number theory and incommensurables
  • Books XI–XIII — solid geometry, closing on the five Platonic solids
  • “There is no royal road to geometry.” — Euclid to Ptolemy
Slide 05

V.The Five Postulates

  • Euclid’s axioms — the foundation stones of plane geometry
  • A straight line may be drawn from any point to any other point.
  • A finite straight line may be extended indefinitely.
  • A circle may be described with any centre and radius.
  • All right angles are equal to one another.
  • If a line crosses two others so that the interior angles on one side sum to less than two right angles, those two lines — if extended — meet on that side. (The parallel postulate.)
  • Four are obvious. The fifth has the air of a theorem in disguise.
Slide 06

VI.The Parallel Problem

  • Two thousand years of trying to prove what cannot be proved
  • The fifth postulate troubled everyone. Where the others fit on a single line, this one needed a paragraph. Surely it was a consequence of the others, not an axiom?
  • Proclus, ibn al-Haytham, Omar Khayyám, Saccheri, Lambert, Legendre — each tried and each failed. Saccheri (1733) thought he had succeeded by deriving an absurdity from its denial. He had not. He had unwittingly proved the first theorems of a new geometry.
  • Equivalent forms
  • Through a point not on a line, there is exactly one line parallel to it.
  • — Or —
  • The angles of a triangle sum to two right angles.
  • Fig. 3 · The fifth postulate, in pictures
Slide 07

VII.The Crack in the Plane

  • Lobachevsky, Bolyai, Gauss · 1820s – 1830s
  • Working independently in Kazan, Pest, and Göttingen, three men dared the unthinkable: deny the fifth postulate and see what follows. The result was not contradiction but a strange, internally consistent world.
  • In hyperbolic geometry, infinitely many lines pass through a point parallel to a given line. Triangles have angles summing to less than 180°. The plane curves away from itself.
  • János Bolyai, 1823
  • “Out of nothing I have created a strange new universe.”
  • Fig. 4 · Poincaré disk model of hyperbolic space
Slide 08

VIII.Riemann’s Manifolds

  • Göttingen, 10 June 1854 · geometry from the inside
  • For his Habilitation lecture, the shy young Bernhard Riemann proposed something that left even Gauss astonished. Geometry, he said, is not about the space figures sit in; it is about a structure intrinsic to the space itself — a way of measuring distance at every point.
  • Drop a sphere, a saddle, a doughnut, or anything else into existence; if you specify a metric — an infinitesimal Pythagorean rule, ds² = gij dxi dxj — you have a geometry. Curvature, distance, angle all follow.
  • Manifold — a space that looks Euclidean up close, anything at large scale
  • Metric tensor — the local rule for measuring length
  • Curvature — how triangles fail to add up to 180°
  • Geometry stops being about a place. It becomes the place.
Slide 09

IX.Gravity is Geometry

  • Einstein, Berlin, November 1915
  • For sixty years Riemann’s manifolds were a mathematician’s curiosity. Then Einstein, struggling to reconcile gravitation with relativity, found in them exactly the language he needed.
  • Mass and energy curve the four-dimensional manifold of spacetime; freely falling bodies trace its straightest possible paths — geodesics. There is no force called gravity. There is geometry, and matter follows it.
  • Einstein field equations · 1915
  • Rμν − ½ gμν R + Λ gμν = (8πG/c4) Tμν
  • Spacetime tells matter how to move; matter tells spacetime how to curve.
Slide 10

X.Topology — Shape Without Size

  • From Euler to Poincaré · geometry stripped to its bare essentials
  • What survives if we forget how to measure? Suppose lines may stretch, surfaces may bend, but nothing tears or fuses. The properties that remain — connectedness, holes, knotting — are topology.
  • Two shapes are equivalent if one can be deformed into the other without cutting. By that rule, a coffee cup is identical to a doughnut: each has exactly one hole.
  • Genus — the count of holes through a surface
  • Euler characteristic — V − E + F, the same for any triangulation
  • Homotopy & homology — algebra detecting shape
  • Fig. 5 · Both have genus 1 — topologically the same surface
Slide 11

XI.Fractals — The Roughness of Things

  • Mandelbrot, IBM Yorktown Heights, 1975
  • Coastlines, clouds, lightning, lungs, river deltas — nature is rarely smooth. Benoît Mandelbrot named the geometry of irregularity fractal: shapes that look the same at every magnification, with non-integer dimension.
  • The Mandelbrot set, born in 1980 of a one-line iteration, became the icon of the field — an infinite, self-similar coastline you can zoom into forever.
  • The iteration
  • zn+1 = zn2 + c (z0 = 0)
  • The set of c for which the sequence stays bounded.
  • Fig. 6 · The Mandelbrot set, with one of its infinite copies
Slide 12

XII.The Modern Workshop

  • Geometry today · many disciplines, one ancient question
  • Where Euclid had a single subject, the modern geometer has a workshop full of them — each a tradition, each in active conversation with physics, computation, and the rest of mathematics.
  • Algebraic geometry — the geometry of polynomial equations; Grothendieck’s schemes; Wiles’s proof of Fermat
  • Differential geometry — smooth manifolds, Lie groups, the language of gauge theory and general relativity
  • Computational geometry — algorithms for meshes, convex hulls, motion planning, the geometry of computer graphics and CAD
  • Discrete & combinatorial — polytopes, sphere packings, the geometry of crystals and codes
  • Geometric analysis — Perelman’s 2003 proof of Poincaré’s conjecture by flowing the metric like heat
  • Three thousand years on, the question is still: what is the shape of things?
Slide 13

XIII.Plates & Pointers

  • Where to read further — and what to watch tonight
  • Books on the shelf
  • Euclid · The Thirteen Books of the Elements (Heath translation)
  • Robin Hartshorne · Geometry: Euclid and Beyond
  • Marvin Greenberg · Euclidean and Non-Euclidean Geometries
  • Michael Spivak · A Comprehensive Introduction to Differential Geometry
  • Benoît Mandelbrot · The Fractal Geometry of Nature
  • Donal O’Shea · The Poincaré Conjecture
  • YouTube searches
  • Pythagorean theorem proof — visual demonstrations
  • Non-Euclidean geometry — hyperbolic and spherical
  • Quick links
  • MacTutor History of Mathematics · St Andrews
  • Mandelbrot set — deep zooms
  • — finis —
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