1. To the 4th Dimension – and beyond!
The Power and Beauty of Geometry
Carlo Heinrich Séquin
University of California, Berkeley
I am Swiss: Italian first name, German middle name, French last name.

2. Basel, Switzerland M N G Math Institute, dating back to 15th century
Granada 2003
Basel, Switzerland
M N G
Math & Science!
Math Institute, dating back to 15th century
I grew up in Basel, CH,
>>> where I attended the M N G (a high school with emphasis in math and science).
>>> During my university years I heard my math lectures in this 500 year old institute, where many famous mathematicians have lectured:

3. Leonhard Euler (1707‒1783) Imaginary Numbers Granada 2003
… such as Leonhard Euler, …

4. Jakob Bernoulli (1654‒1705) Logarithmic Spiral Granada 2003
or Jakob Bernoulli, the inventor od the logarithmic spiral.
From an early age on, I was fascinated with numbers and geometry.

5. In 11th Grade: Descriptive Geometry
ISAMA 2004
In 11th Grade: Descriptive Geometry
In 11th grade we had a subject called: Descriptive Geometry, where we constructed the intersections between cylinders and cones – without computers!
I thought that this was very cool stuff!

6. Geometry in Every Assignment . . .
Granada 2003
Geometry in Every Assignment . . .
CCD TV Camera RISC 1 Micro Chip Soda Hall
So, in many of my future jobs I was focusing on the geometry component: Laying out an integrated circuit chip is one big geometry problem!
On the left is the first solid-state TV camera based on Charge-Coupled Devices that we built at Bell Labs. In the middle is the Reduced Instruction Set Microcomputer that I developed with my colleague Dave Patterson at UC Berkeley and our graduate students. Then we needed a new building for Computer Science, and I switched from 2-dimensional geometry to 3D. >>> More recently I created programs to help artists do geometrical sculptures. And I also created Math visualization models: This is the topic of today’s talk.
“Pax Mundi” Hyper-Cube Klein Bottle

7. 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.
Why is geometry so useful in so many fields? -- It is a power tool for seeing patterns.
And discovering patterns is a basis for true understanding.
For my sculptures, I find such patterns in inspirational artwork by more intuitive artists, and I then capture them in the form of a computer program.
=== But “seeing patterns” really happens in your mind!!

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 ?
In this talk I will introduce you to a way of seeing things in your mind that cannot be seen in an ordinary way.
The left figure looks like 6 triangles, or a hexagon with some internal lines. -- But if we rotate this figure a little, it suddenly starts to look like a 3D cube.
This is a simple example of an image that is composed by your mind.

9. Seeing a Mathematical Object
Very big point
Large point
Small point
Tiny point
Mathematical point
Here is how our high school teacher introduced us to seeing a mathematical object: Here is a very big point, and I am sure that everyone can see it.
>>> Next is a point that is still fairly large – Can you still see it ?
>>> Small point: WHO can still see this ?
>>> Tiny point: …
>>> Mathematical point: … use your mind! If you can see it, you are doing mathematical seeing!

10. Geometrical Dimensions
Point Line Square Cube Hypercube D D D D D D
EXTRUSION
Now we use this “mathematical seeing” to go from a simple point to a 4D hypercube by a process called EXTRUSION:
For this, I put on my magical extrusion gloves … PANTOMIME :
We start with a single mathematical point – which is infinitely small, so you cannot really see it, but you know where it is.
--- Then we stretch this point into a thin mathematical line …
===
--- The last step is difficult for us to comprehend. Perhaps we can understand why this is -- with an analogy to FLATLAND…

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:
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 – even thought they cannot see them the way you can; they can only walk around them and measure the lengths of sides and the angles between them, and make sure that they are all the same. -- Then they can “see” them in their minds.
3-gon gon gon gon 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!
One day, a visitor comes from our 3D world and tries to explain to them what a CUBE is: She says: Just take a square and extrude it upwards,
-- NO, not in a diagonal direction, but truly “UPWARDS”, in a direction that is perpendicular to both edge directions of the square. This is difficult to do for the flat worms!!
Same as for us, when we try to see a HYPER-CUBE. Perhaps, some 4D aliens looking down on us also smile at our efforts trying to point into the 4th dimension!
But it is out there !!

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?
OK – Let’s step back and look at some beautiful geometry that we _can_ see: the Platonic Solids! They have been known to mankind for 2 millennia.
“Regular” means that all faces, edges, and corners are indistinguishable; they are all exactly the same.
The surfaces of these objects are composed of regular 2D polygons. There were infinitely many of those regular 2D n-gons.
But there are only five regular polyhedra in 3 dimensions. – Do you know why? Could you prove this to your friends -- or to your grandmother? -- Soon you will be able to do this!

14. Making a Corner for a Platonic Solid
Put at least 3 polygons around a shared vertex to form a real physical 3D corner!
Putting 3 squares around a vertex leaves a large (90º) gap;
Forcefully closing this gap makes the structure pop out into 3D space, forming the corner of a cube.
We can also do this with 3 pentagons:  dodecahedron.
In order to construct the surface of some Platonic solid, we have to assemble some 2D polygons so that they form real physical 3D corners; (CARDBOARD MODEL).
Thus we need at least 3 faces coming together at each corner. When we put 3 squares around a shared vertex, then there is still a large gap left. Now we forcefully close this gap, and the structure pops out into the 3rd dimension, and a real cube corner is formed. 8 of these make a complete cube.
We can also use 3 pentagons, and the result will be the dodecahedron.

15. Why Only 5 Platonic Solids?
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
Now we can see why there are only 5 Platonic solids: When we use triangles for the surface facets, we have a choice, we can use 3 or 4 or 5 around a point (use fingers) to form a regular pyramid-like corner. This leads to the Tetrahedron, the Octahedron, and the Icosahedron which uses 20 triangles. …
But if we use squares, there is only one option: 3 squares around a corner – forming a cube.
And we have already seen that with 3 pentagons we can make a dodecahedron.
All larger n-gons are to “round” and are not able to make true 3D corners.

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!
Now we take a leap into the 4th dimension! We apply exactly the same kind of thinking to answer the question: “How many …”
=== WHAT DO YOU THINK? -- WHO predicts: Same as in 3D - just 5? -- More than in 3D? -- Fewer than in 3D?
>>> Let’s find out: The key insight is: The “crust” of the regular 4D Polytopes is made of Platonic solids; so we have to see how we can build viable 4D corners from a few of these solids.

17. Constructing a 4D Corner:
Forcing closure: ? 3D 4D
Let’s do again what we did for the construction of Platonic solids: We took a few regular n-gons that did not fill the space around a vertex, and forced the gap to close.
Now we do this again, one dimension higher! We will take a few Platonic solids, say cubes, and fit them around a common edge (shown in red). I use the same 2D model again, but now assume that a 3D cube is behind each square!
Once again, we force the left-over wedge of empty space to close, and POP – we get a 4D corner!
creates a 3D corner creates a 4D corner

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°.
Now we have to count in how many different ways we can do this. The crucial value now is the dihedral angle, i.e., the angle between two adjacent faces of a Platonic cell; it tells us how many of a particular solid can be fit around a shared edge, -- and how much space is left over! For the tetrahedron this critical angle is about 70 degrees,
>>> so we can fit 3, 4, or 5 tetrahedra around a common edge, and still have a little room to do some bending.

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°).
… so we can fit 3, 4, or 5 tetrahedra around a common edge, and still have a little room to do some bending.
and all these options yield valid regular 4D polytopes: specifically: the simplex, the cross polytope, and the 600 cell.
With cubes there is only one valid option: 3 cubes around an edge form a hypercube corner.
We can also use 3 octahedra to make a valid corner, or 3 dodecahedra.
But the icosahedron is useless: its interior angle at each edge is more then 120 degrees, 3 of them don't fit around a common edge. So we know what is possibel in some abstract manner – But can we get an idea what these things might look like? Let’s use some “mathematical seeing.”

20. Wire-Frame Projections
Project 4D 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.
We cannot directly look into 4-D space. But we can make projections of these 4D objects down to 3D.
For instance we could cast shadows – this one kind of projection. But this is not very informative, it just gives feature-less blobs!
It is much better to project wireframes of these objects, -- then we can see through them, and also see their backsides.
The figure on the right is only a 2D image, but your brain allows you to see it as a cube – a 3D object.
With a little practice you can also look at 3D projections and gain an understanding of the 4D object from which it was derived.

21. Oblique or Perspective Projections
3D Cube  2D D Cube  3D ( 2D )
We have some further options of how we want to do this projection – just as in engineering where we can use either an oblique or a perspective projection to show an ordinary 3D cube.
== In addition, we can also use color to give depth information: Here, red is the front object, blue is the extruded back object, and green are the transition edges going from front to back.
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)
Now we start the parade of the six regular 4D polytopes:
First is a perspective projection of the 4D Simplex. It is composed of 5 tetrahedra.
In 4D all 5 corners are at equal distance from each other, and all edges are of the same length.
They look different here, because I used a perspective projection.

23. 16-Cell or “4D-Cross Polytope”
16 cells (tetrahedra),
32 faces,
24 edges,
8 vertices.
If we use 4 tetrahedra around a shared edge, we get the 16 cell , also called the 4D Cross polytope. It has a total of 16 tetrahedral cells, but only 8 vertices.
It is the geometric dual of …

24. 4D-Hypercube or “Tessaract”
8 cells (cubes),
24 faces (squares),
32 edges,
16 vertices.
The Hypercube or Tessaract, which as 8 cubic cells and 16 vertices.
Here I used a perspective projection with depth-color information.

25. 24-Cell 24 cells (octahedra), 96 faces, 96 edges, 24 vertices.
1152 symmetries!
This is my favorite: 24 octahedra make the beautiful 24-Cell. It is self-dua -- lthis means, it has the same number of cells and vertices.
It has 1152 symmetries! – but unfortunately, you can only see them in 4D space!

26. 120-Cell 120 cells (dodecahedra), 720 faces (pentagons), 1200 edges,
600 vertices.
Aligned parallel projection (showing less than half of all the edges.)
Finally there are two big ones! Here is the 120 cell -- made of 120 dodecahedra. It has 1200 edges. But in this parallel projection you can only see about half of them, because some edges obscure some other ones.

27. (smallest ?) 120-Cell
Wax model, made on a Sanders RP machine (about 2 inches).
Here is a perspective projection of the same thing. In principle it shows all 1200 edges, but some of them lie close to the center of this model and are very small!

28. 600-Cell 600 cells, 1200 faces, 720 edges, 120 vertices.
Parallel projection (showing less than half of all the edges.)
By David Richter
Last but not least: a parallel projection of the 600 cell, containing 600 tetrahedra.
This model was made by David Richter – I believe from drinking straws.

29. Beyond 4 Dimensions … What happens in higher dimensions ?
Only THREE for each dimension!
Pictures for 6D space:
Simplex with 6+1 vertices,
Hypercube with 26 vertices,
Cross-Polytope with 2*6 vertices.
Beyond 4 Dimensions …
What happens in higher dimensions ?
How many regular polytopes are there in 5, 6, 7, … dimensions ?
The constructive process can be continued. We can use the 4-dimensional regular polytopes to form the surfaces of the 5-dimensional polytopes.
Doing the same kind of analysis as before, it is not too difficult to show that there are only three such regular polytopes in all dimensions higher than 4.
Here I show some projections for the 6-dimensional regular polytopes.
>>> But now I want to use the last three minutes to introduce you to some other strange and beautiful geometrical objects. . .

30. Bending a Strip in 2D The same side always points upwards.
The strip cannot cross itself or flip.
I have focused on polyhedral objects to give you an introduction to higher dimensions. But higher dimensions are even more fun when we create smooth, free-form shapes.
Let me give you a glimpse of the added capabilities we obtain when we go to higher dimensions:
In 2 dimensions, a strip of material can bend and curve, but it cannot cross itself, and no matter how badly we distort it, it can never fip over; the same green side will always be pointing upwards.

31. Bending a Strip in 3D
Strip: front/back Annulus or Cylinder Möbius band
Twisted !
But in 3D space, we can do more interesting things with a strip of paper: We can either close it into a cylinder, OR --
we can twist it so that the colorful front-side meets the B&W back-side! Now this loop is single-sided! – it is a Moebius band! This is a fascinating shape:
Take a strip of paper and some scotch tape and glue the ends together in this twisted manner; -- then cut it down the middle along the B&W line ... What do you think you will get? -- You have to try that yourself !
This Moebius band also forms a crucial element in many intriguing geometrical sculptures…

32. 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)
Here are some examples: “Aurora Borrealis” is just one, long, wound-up Moebius band.
“Tripartite Unity” by Max Bill is of genus 3, that means it is a made of three Moebius bands.
The “Minimal Trefoil” is a composite of four Moebius bands.
The “Heptoroid” is really complicated; it is the equivalent of 22 Moebius bands!
>>> All these sculptures have just one single edge. -- More complex stuff is possible with multiple edges! But, is it possible to have single-sided surface with NO edges at all ?? (Like a sphere or a donut, which are surfaces without any edges; but they are two-sided: They have an outside in a color that you can see, and an inside in a color that you cannot see.)

33. Single-Sided Surfaces Without Edges
Boy-Surface Klein Bottle “Octa-Boy”
YES it is possible, and on the left are the two simplest cases: The Boy-surface and the Klein bottle.
On the right is something that I recently constructed by glueing together the edges of eight Moebius bands.
All of these 3D models have self-intersections. To see these surfaces cleanly without self-intersections, we have to go to 4D space!
The Klein bottle (in the center) is probably the best known object of this kind. . .

34. More Klein Bottle Models
Lot’s of intriguing shapes!
It comes in many different shapes! On the left is Cliff Stoll from Berkeley, presenting one of his large classical glass Klein bottles.
But all the shapes on the right are also Klein bottles -- where the inside is always smoothly connected to the outside.
Some make for cool sculptures and possibly even good climbing structures!

35. The Classical Klein Bottle
Every Klein-bottle can be cut into two Möbius bands! =
The last cool fact that I want you to know: Every Klein-bottle can be cut into exactly TWO Moebius bands!
Here I show you how this is done for the classical shape of a Klein bottle. -- And here is a more twisted MODEL in which the resulting Moebius bands are interlinked.
These intriguing shapes are the topic for a separate talk at some other time.

36. Conclusion Geometry is a powerful tool for S & E.
It also offers much beauty and fun!
(The secret to a happy life … )
In conclusion: Geometry is a powerful tool for almost all fields of science and engineering.
It also offers much beauty and fun!
The latter is very important to me, because I believe that the secret to a happy life is – to find out what you really like to do -- and then find somebody to pay you to do it.
I have been in that mode for the last 30 years of my life! – And I hope that you will be bale to do that too!
Thanks! – Questions ?

37. What is this good for?
Klein-bottle bottle opener by Bathsheba Grossman.
Mathematicians create new insights that are often used only many years later.
Are there practical applications of what I have shown you just now ??
On the web you can find this bottle opener in the shape of a Klein bottle …
Also, in Bristol, England, they are building a Moebius bridge!
I am not aware of any architect who has built a Klein-bottle house yet. – But there is a YouTube video that shows some models.