To the 4 th Dimension and beyond The

To the 4 th Dimension – and beyond! The Power and Beauty of Geometry Carlo Heinrich Séquin University of California, Berkeley

Basel, Switzerland MNG Math Institute, dating back to 15 th century Math & Science!

Leonhard Euler (1707‒ 1783) Imaginary Numbers

Jakob Bernoulli (1654‒ 1705) Logarithmic Spiral

In 11 th Grade: Descriptive Geometry

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

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

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

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

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

Flat-Land Analogy • Assume there is a plane with 2 D “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. . .

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!

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

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

Why Only 5 Platonic Solids? 4 T 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! 8 T 20 T

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

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

How Do We Find All 4 D 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°.

All Regular Polytopes in 4 D 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°).

Wire-Frame Projections • Project 4 D polytope from 4 D space to 3 D 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.

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

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

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

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

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

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

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

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

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

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

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

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