# Researcher builds four-dimensional figure in his office

##### March 17, 2015 by Diana Lutz, Washington University in St. Louis

The curious child of bookish parents who browses through their libraries sometimes falls down a rabbit hole while apparently reading quietly in an armchair.

Something like this happened to Ivan Horozov, PhD, the William Chauvenet lecturer in the Department of Mathematics in Arts & Sciences at Washington University in St. Louis. Horozov grew up in Sofia, Bulgaria, the child of a mathematician whose library included the book "Geometry and the Imagination" by Hilbert and Cohn-Vossen.

The book, which he says reads like a storybook rather than a textbook, explores regular geometric shapes—which mathematicians call polytopes—in two, three and four .

A square is a two-dimensional polytope; its three dimensional analog is a cube and its four-dimensional analog is something called a tesseract.

The book, of course, could only show two-dimensional projections of the three- or four-dimensional objects it described. For example, on the page the drawing of a cube was a square within a square.

To make three-dimensional models of the three-dimensional polytopes, Horozov constructed their polygonal faces with a straightedge and compass, cut them out and glued them together along their edges. In this way he made all five three-dimensional polytopes (the tetrahedron, cube, octahedron, dodecahedron and icosahedron).

Then he started on the four-dimensional polytopes. There are six of those, he learned.

"The book illustrated the four-dimensional analog of a tetrahedron, the four-dimensional analog of a cube, the four-dimensional analog of an octahedron and a four-dimensional poltyope that has no analog in other dimensions," Horozov said.

"The remaining two, it said, were too complicated to draw. So I decided if I could not draw them, I would at least imagine what the polytopes should look like.

It took me two years, but by the time I was 12 I could imagine the fifth four-dimensional polytope, which is made up of 120 dodecahedrons."

But he couldn't build it until he arrived in St. Louis to take a postdoctoral teaching position. Here he discovered Zometools, construction sets invented by a designer who was interested in polygonal buildings, like Buckminister Fuller's dome.

Horozov "assimilated" all of the Zomes in the department and began to build the 120-cell polytope. The finished model, a giant, gaily colored, weirdly distorted cobweb of balls and struts now occupies most of his office. It is just possible to close the door without brushing against it.

"Of course you can't throw away a dimension without losing information. The projection of a cube onto a sheet of paper is a square within a square, but the two squares are now ever so slightly distorted," Horozov said. In the case of the 120-cell polytope, the innermost dodecahedron is regular, but as you move outward form the center the dodecahedrons are more and more deformed.

Horozov is now working on the sixth four-dimensional polytope, which is, he said, the dual of the 120-cell polytope. A dual is the figure that emerges when the vertices of one figure become the faces of another and the faces the vertices. The dual of a cube, for example, is an octahedron. And the dual of the polytope made of 120 dodecahedrons is a polytope made of 600 tetrahedrons.

Even the Greeks knew about the three-dimensional polytopes, often called Platonic solids. But the four-dimensional polytopes were discovered only in the 19th century and even then the mathematician who described them was unable to get his work published because it was considered so outlandish.

After all it isn't intuitive that there might be more than three spatial dimensions. Perhaps this is why magical properties have been ascribed to the four-dimensional polytopes. In Madeleine d'Engle's "Wrinkle in Time," for example, the tesseract is a time-travel device that wrinkles up the space-time continuum and creates a passage from one part of it to another.

Horozov hasn't read "A Wrinkle in Time," but he does follow string theory, theories in physics that predict there are extra dimensions. One version, for example, predicts spacetime has 11 dimensions and that we just don't see anything except the three spatial dimensions and time because the extra dimensions are "compactified," or rolled up tight.

String theory, like the four-dimensional polytopes, is still very much an imaginative exercise, but should experimental evidence of its validity ever surface, we will all have to cultivate multi-dimensional thinking.

Explore further: Platonic solids generate their 4-dimensional analogues

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##### LariAnn
3.7 / 5 (3) Mar 17, 2015
I find that the apparent challenge with "higher" dimensions lies in an apparent misunderstanding of what a "dimension" really is. Basically, a dimension is a parameter that defines an object in space-time. The first three dimensions are obvious as they, together, define a volume in space, but they do not define an object in space because they do not provide anywhere near enough information about the object. Therefore, we need a lot more dimensions to do this. It doesn't require esoteric mathematics, but a knowledge of fractal geometry helps. For example, in space-time, objects are not static - they are dynamic. So there have to be dimensions that define movement in space-time. Imagine a line of infinite length, rotating on a point axis. The frequency of the rotation is a dimension. Same for an infinite plane rotating on a line axis. See how easy it is to find new dimensions?
##### Noumenon
5 / 5 (2) Mar 17, 2015
There are "Physical dimensions" and then there "Mathematical dimensions". The reality of 'Physical dimensions' are usually observationally quantified in some way by say energy distributed over degrees of freedom and conservation laws,.... while 'Mathematical dimensions' are merely particular to the model and circumstance of what is being modeled, just variables needed,... like in configuration space, or phase space, or Fock space.
##### jerry_bushman_7
2.6 / 5 (5) Mar 17, 2015
What a bunch of whack doodles. Any observable object has 3 and only 3 dimensions. A 4dimesional object in his office? Bull crap. This is not about fantasy and the twisted way you can manipulate understanding. Truth is there is only 3 dimensions unless you are psychotic.
##### harmonograms
5 / 5 (2) Mar 17, 2015
The polyhedra book shown in the video was Cundy and Rollett's "Mathematical Models", an indispensable reference for physicists. It's currently out of print but still widely available if you look around. I construct model tesseracts out of brass tubing and elastic cord, making the vertices look elegant is a problem; I also attended several Buckminster Fuller lectures back in the day...
##### RobertKarlStonjek
1 / 5 (2) Mar 17, 2015
Each dimension can be reduced to the next lower dimension by removing the higher dimensional property or by removing any one of the dimensions.

A two dimensional object can be reduced to one dimensional lines by removing the surface property or by removing either height or width resulting in a series of lengths.

Lengths can be reduced to zero dimensional points by removing the magnitude property or the remaining length/height/depth dimension.

Three dimensional objects can be reduced to surfaces if the volume property is removed or to two of the three dimensions.

For the tesseract this can't be done. The fourth dimension is added to an analogy of three dimensions (x,y and z axes) rather than being derived from reality.

Using time as a fourth dimension we note that removing the time property results in a series of three dimensional objects OR by removing one of the four dimensions we arrive at three dimensions eg a moving surface or a solid frozen object.
##### felicien_perrinn
not rated yet Mar 18, 2015
Dimensions exist only in the human skull: it's called "preceding intemporal inflexive contiguity"!
##### antialias_physorg
5 / 5 (1) Mar 18, 2015
I find that the apparent challenge with "higher" dimensions lies in an apparent misunderstanding of what a "dimension" really is.

They are specifically talking about an object with four spatial dimensions. Yes, it's quite easy to imagine an n-dimensional object if you are free to choose what kind of dimensions to use. A dimension is just anything that is independent of any of the other dimensions - i.e. something you can change in any way without affecting any of the others.
That is why space-time is a bit tricky in that regard, because when you move to high speeds it actually isn't 3D space and 1D time. I.e. it's not 4 dimensions because time and space are interconnected (time dilation, relativistic compression, ...). This is why it's called "space-time" and not "space and time".

If you stipulate four spatial dimensions from the outset, however, then the brain runs into trouble.
##### scott_c_waring
not rated yet Mar 18, 2015
This is a three dimensional object. Let me explain. If you made this object from a plastic bag you would see it, but that bag would have to be filled with water to make similar.
1. Create an airtight plastic bag that is long 1foot by 3 inches.
2. Create a tunnel through this bag (1 inch will suffice), from end to end long ways. The tunnel has to allow air through it.
3. fill the plastic bag with liquid and then seal it it.

Now when you hold onto the back it will turn itself inside out...in the exact manner seen in this 4D object above, except it really 3D. This object above is just flipping around on itself, like a Childs toy. But it sure is cool to look at. To bad I can't upload photos here, I could take a picture of this childs toy on my sons desk. Its the same.
##### Noumenon
5 / 5 (1) Mar 18, 2015
@scott,... Keep in mind that the above image is just a "two-dimensional projection of the four-dimensional " object, so you're not seeing the whole thing.
##### NathanLarkinCoppedge
not rated yet Mar 18, 2015
Check out the Dimensional Encyclopedia. It may not deal with Buckminster balls, but at least it connects category theory with philosophy and other subjects. It is a work in process, but the first three volumes on Philosophy, Psychology, and Biology have been published.
##### Dethe
not rated yet Mar 18, 2015
The objects around us are mostly highly hyperdimensional surfaces. You should realize, that the surface-volume ratio increases with number of dimensions and that we can see only 3D slice of these objects. The results looks pretty similar like the common atoms hold with invisible forces at distance. (illustration)
##### viko_mx
1 / 5 (2) Mar 22, 2015
Time is not forth dimension. We can not unite 3 spatial dimensions with the speed with which are changed coordinates and states of the objects located in the first 3 dimensions. Therefore the expression space-time has no physical sence.
##### Shootist
5 / 5 (1) Mar 22, 2015
"And he built a crooked house" - Robert A. Heinlein.

Been there, done that.
##### Mike_Massen
3 / 5 (2) Apr 04, 2015
viko_mx claimed
Time is not forth dimension. We can not unite 3 spatial dimensions with the speed with which are changed coordinates and states of the objects located in the first 3 dimensions. Therefore the expression space-time has no physical sence.
Well obviously because you are ignorant of Einstein's work and Jesus NEVER touched on maths & neither did Moses !

Please get an education, your claimed god is so very far behind, he doesn't communicate well !
##### viko_mx
1 / 5 (1) Apr 04, 2015
You probably agree that not all people have to accept the theory of relativity for reliable and give it mythical image, as do many of the world scentifically (conditionaly) oriented media. It is known how is created legends. Its popularity is due to the fact that denied Creation that were recorded in Bible and support materialistic wolrdview with introduction of the fictional physical phenomena as elastic space, which gives the green light to the theory of cosmic evolution known as Big Bang theory. This theory introduces abstract concepts such as space - time that are purely mathematical speculation that have nothing to do with reality. You may not know, but such mathematical speculations not always reflect the reality and can describe completely random fabulous objects.
##### viko_mx
1 / 5 (1) Apr 04, 2015
All real objects and interactions in the universe probably have a mathematical description, but not all mathematical objects have manifestation in reality.