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To the 4 th Dimension – and beyond!

The Power and Beauty of Geometry. To the 4 th Dimension – and beyond!. Carlo Heinrich Séquin University of California, Berkeley. Basel, Switzerland. M N G. Math Institute, dating back to 15 th century. Math & Science!. Leonhard Euler (1707‒1783). Imaginary Numbers. Logarithmic Spiral.

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To the 4 th Dimension – and beyond!

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  1. The Power and Beauty of Geometry To the 4th Dimension – and beyond! CarloHeinrichSéquin University of California, Berkeley

  2. Basel, Switzerland M N G Math Institute, dating back to 15th century Math & Science!

  3. Leonhard Euler (1707‒1783) Imaginary Numbers

  4. Logarithmic Spiral Jakob Bernoulli (1654‒1705)

  5. In 11th Grade: Descriptive Geometry

  6. Geometry in Every Assignment . . . CCD TV Camera RISC 1 Micro Chip Soda Hall “Pax Mundi” Hyper-Cube Klein Bottle

  7. Geometry • A “power tool” for seeing patterns. • Patternsare a basis for understanding. • For my sculptures, I find patterns in inspirational art work and capture them in the form of a computer program.

  8. Mathematical “Seeing” • See things with your mind that cannot be seen with your eyes alone. A hexagon plus some lines ? or a 3D cube ?

  9. Seeing a Mathematical Object • Very big point • Large point • Small point • Tiny point • Mathematical point

  10. Geometrical Dimensions Point - Line - Square - Cube - Hypercube - ... 0D 1D 2D 3D 4D 5D EXTRUSION

  11. Flat-Land Analogy • Assume there is a plane with 2D “Flat-worms” bound to live in this plane. • They can move around, but not cross other things. • They know about regular polygons: 3-gon 4-gon 5-gon 6-gon 7-gon . . .

  12. Explain a Cube to a Flat-lander! • Just take a square and extrude it “upwards” . . .(perpendicular to both edge-directions) . . . • Flat-landers cannot really “see” this!

  13. The (regular, 3D) Platonic Solids • All faces, all edges, all corners, are the same. • They are composed of regular 2D polygons: Tetrahedron Octahedron Cube Icosahedron Dodecahedron • There were infinitely many 2D n-gons! • How many of these regular 3D solids are there?

  14. Making a Corner for a Platonic Solid Put at least 3 polygons around a shared vertexto form a real physical 3D corner! • Putting 3 squares around a vertexleaves a large (90º) gap; • Forcefully closing this gapmakes the structure pop out into 3D space,forming the corner of a cube. • We can also do this with 3 pentagons: dodecahedron.

  15. Why Only 5 Platonic Solids? 4T Lets try to build all possible ones: • from triangles: 3, 4, or 5 around a corner: • from squares: only 3 around a corner: • from pentagons: only 3 around a corner: • from hexagons:  “floor tiling”,does not bend! • higher n-gons:  do not fit around a vertex without undulations (forming saddles);  then the edges would no longer be all alike! 8T 20T

  16. The “Test” !!! How many regular “Platonic” polytopes are there in 4D ? Their “surfaces” (= “crust”?) are made of all regular Platonic solids; and we have to build viable 4D corners from these solids!

  17. Constructing a 4D Corner: creates a 3D corner creates a 4D corner 3D 2D Forcing closure: ? 4D 3D

  18. How Do We Find All 4D Polytopes? • Reasoning by analogy helps a lot:-- How did we find all the Platonic solids? • Now: Use the Platonic solids as “tiles” and ask: • What can we build from tetrahedra? • or from cubes? • or from the other 3 Platonic solids? • Need to look at dihedral angles: Tetrahedron: 70.5°, Octahedron: 109.5°, Cube: 90°, Dodecahedron: 116.5°, Icosahedron: 138.2°.

  19. All Regular Polytopes in 4D Using Tetrahedra (70.5°): 3 around an edge (211.5°)  (5 cells) Simplex 4 around an edge (282.0°)  (16 cells) Cross-Polytope 5 around an edge (352.5°)  (600 cells) 600-Cell Using Cubes (90°): 3 around an edge (270.0°)  (8 cells) Hypercube Using Octahedra (109.5°): 3 around an edge (328.5°)  (24 cells) 24-Cell Using Dodecahedra (116.5°): 3 around an edge (349.5°)  (120 cells) 120-Cell Using Icosahedra (138.2°): None! : dihedral angle is too large ( 414.6°).

  20. Wire-Frame Projections • Project4D polytope from 4D space to 3D space: • Shadow of a solid object is mostly a “blob”. • Better to use wire frame, so we can also see what is going on at the back side.

  21. Oblique or Perspective Projections 3D Cube 2D 4D Cube  3D ( 2D ) We may use color to give “depth” information.

  22. 5-Cell or “4D Simplex” • 5 cells(tetrahedra), • 10 faces (triangles), • 10 edges, • 5 vertices. (Perspective projection)

  23. 16-Cell or “4D-Cross Polytope” • 16 cells (tetrahedra), • 32 faces, • 24 edges, • 8 vertices.

  24. 4D-Hypercube or “Tessaract” • 8 cells (cubes), • 24 faces (squares), • 32 edges, • 16 vertices.

  25. 24-Cell • 24 cells(octahedra), • 96 faces, • 96 edges, • 24 vertices. 1152 symmetries!

  26. 120-Cell • 120 cells (dodecahedra), • 720 faces (pentagons), • 1200 edges, • 600 vertices. Aligned parallel projection(showing less than half of all the edges.)

  27. (smallest ?) 120-Cell • Wax model, made on a Sanders RP machine (about 2 inches).

  28. 600-Cell • 600 cells, • 1200 faces, • 720 edges, • 120 vertices. Parallel projection(showing less than half of all the edges.) By David Richter

  29. Beyond 4 Dimensions … • What happens in higher dimensions ? • How many regular polytopes are therein 5, 6, 7, … dimensions ? • Only THREE for each dimension! • Pictures for 6D space: • Simplex with 6+1 vertices, • Hypercube with 26vertices, • Cross-Polytope with 2*6 vertices.

  30. Bending a Strip in 2D • The same side always points upwards. • The strip cannot cross itself or flip.

  31. Bending a Strip in 3D Strip: front/back Annulus or Cylinder Möbius band Twisted !

  32. Art using Single-Sided Surfaces All sculptures have just one continuous edge. Aurora Borrealis C.H. Séquin(1 MB) Tripartite UnityMax Bill(3 MB) Minimal Trefoil C.H. Séquin(4 MB) Heptoroid Brent Collins(22 MB)

  33. Single-Sided Surfaces Without Edges Boy-Surface Klein Bottle “Octa-Boy”

  34. More Klein Bottle Models Lot’s of intriguing shapes!

  35. The Classical Klein Bottle • Every Klein-bottle can be cut into two Möbiusbands! = +

  36. Conclusion • Geometry is a powerful tool for S & E. • It also offers much beauty and fun! • (The secret to a happy life … )

  37. What is this good for? • Klein-bottle bottle opener by Bathsheba Grossman.

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