1 / 37

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.

albin
Download Presentation

To the 4 th Dimension – and beyond!

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related