A Geometry Thing That’s Left Me Unsettled


I came across a little geometry thing that left me unsettled, even as I have to admit it’s correct. Start with a two-dimensional space, or as the hew-mons call it, a plane. Draw a square with sides of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1), (+1, -1), (-1, +1), and (-1, -1). Around each of these corners draw a circle of radius 1.

There is some largest circle that you can draw, centered on the origin, the dead center of the square, with Cartesian coordinates (0, 0), and that just touches all of the corners’ circles. It has a radius of a little under 0.414.

Now think of the three-dimensional analog. Three-dimensional space. Draw a box with sides all of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1), (+1, +1, -1), (+1, -1, +1), (+1, -1, -1), (-1, +1, +1), (-1, +1, -1), (-1, -1, +1), and (-1, -1, -1). Around each of these eight corners draw a circle of radius 1.

There is some largest sphere that you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0), that just touches all of the corners’ circles. It has a radius of a little under 0.732.

Think of the four-dimensional analog. This is harder to sketch. But a four-dimensional hypercube, with each side of length 2 and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1, +1), (+1, +1, +1, -1), (+1, +1, -1, +1), (+1, +1, -1, -1), and you know what? Will you let me pretend we listed all sixteen corners? Thanks. Around each of these corners draw a circle of radius 1.

There is some largest hypersphere you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0, 0), and that just touches all of these corners’ circles. It has a radius of 1.

Keep going. Five-dimensional space, with corners like (+1, +1, +1, +1, +1). Six-dimensional space, with corners like (+1, +1, +1, +1, +1, +1). Seven-dimensional space. And so on.

Eventually, the space is vast enough that the radius of this largest-touching hypersphere is bigger than 2. That is, reaching out more than twice as far as the original box goes, this even though the corner hyperspheres line the edges of the box, and touch one another along those edges.

Non-Euclidean geometry has the reputation of holding deep, inscrutable mysteries. To say something is a non-Euclidean space, outside of a mathematical context, is to designate it as a place immune to reason and beyond human comprehension. This is not such a case. This is a completely Euclidean space; it’s just got a lot of dimensions to it. Strange things will happen.

Another weird, but to me not so unsettling matter, concerns the surface (or hypersurface) area and the volume of these spheres. The circumference of a unit circle is, famously, 2π. The area of a unit sphere is 4π. For a four-dimensional hypersphere the surface area is a bit bigger yet. And bigger again for five and six and seven dimensions. But at eight dimensions the surface area starts shrinking again, and it never grows again. Have a great enough number of dimensions and the unit hypersphere has almost zero surface area. The volume of a unit circle is π. Of a unit sphere, \frac43 \pi . For a four-dimensional hypersphere, \frac12 \pi^2 . For a five-dimensional hypersphere, \frac{8}{15}\pi^2 . It is never so large again; for six or more dimensions the volume starts to shrink again. As the number of dimensions of space grows, the volume of the unit hypersphere dwindles to zero.

You know, that’s unsettling me more now that I’m paying attention to it.

By the way, Thimble Theatre is trying to explain the fourth dimension


I hope to have proper comment about it in the usual Sunday Reading the Comics post. But the “current” storyline in Elzie Segar’s Thimble Theatre comic strip — Popeye to normal people — is the 1936 introduction of Eugene the Jeep. If you’ve looked at my user icon here you know I like Eugene.

Anyway, Eugene the Jeep has wondrous powers. These include the power of prophecy and the power to disappear from even enclosed spaces. Segar’s explanation for this was that the Jeep can turn into the fourth dimension and so do things we can’t hope to do. Which is a fun premise, yes. More, though, it’s got to be a pretty early use of the fourth or other high dimensions in pop culture. Yes, there were some things normal people might know that talk about higher dimensions. H G Wells’s The Time Machine starts with talk about time as a dimension like space. Edwin Abbott’s Flatland is explicitly about two- and three-dimensions, although Square thinks of whether there could be four- or more-dimensional spaces.

Professor: 'I am Professor Gipf, solver of scientific mysteries. Have you a mystery to be solved?' Olive Oyl: 'I have! We locked this Jeep in a room and he escaped. He is also able to disappear right before my eyes.' Professor: 'Is Zasso? I suspect he has fourth-dimensional qualities. It's lucky I have my dimension detector.' (He pulls out many bundles of wires, and more and more, making a huger tangle.) Professor: 'Darn it! Oh, darn the luck!' Olive Oyl: 'Is something missing?' Professor: 'Yes. Have you got a piece of wire?'
Elzie Segar’s Thimble Theatre (Popeye) for the 25th of October, 2019. It originally ran the 26th of May, 1936. Fun fact: that last panel shows the extension strip in the living room where we have the TV, DVR, Blu-Ray, Switch, and record player plugged in.

Wikipedia helps me find a few pieces of literature mentioning the fourth dimension before Eugene the Jeep. And a few pieces of visual art as well. No mention of earlier comic strips, although there’s no mention of Eugene the Jeep in either. So, all I can say is this is an early pop cultural appearance of the fourth dimension. I can’t say it’s the first, even among major comic strips.

Do not try to use this to pass your geometry quals.

Slowly Rotating Hyperdodecahedron


Here’s an engaging moving picture from RobertLovesPi. The Platonic solids — cubes, pyramids, octahedrons, icosahedrons, and dodecahedrons — are five solid shapes each with the same regular convex polygon as their face. This is a nice two-dimensional rendering of a three-dimensional projection of a “hyperdodecahedron”. It’s made of 120 dodecahedrons, in a four-dimensional space. And it’s got the same kind of structure that Platonic solids have, being made of the same regular convex polyhedron for each face.

Remarkably, I learn from Mathworld, the shape is three-colorable. That is, suppose you wanted to assign colors to each of the corners in this four-dimensional shape. They’re all green circles here, but they don’t have to be. There are a lot of these corners, and they’re connected in complicated ways to one another. But you could color in every one of them, so that none if them is connected directly to another of the same color, using only three different colors.

RobertLovesPi.net

This is the hyperdodecahedron, or 120-cell, one of the six four-dimensional analogs of the Platonic solids. It’s been shown on this blog before, but this image has one major change: a much slower rotational speed. It is my hope that this will help people, including myself, with the difficult task of understanding four-dimensional objects.

5-Hi, 120-cell, Hecatonicosachoron

This image was created using Stella 4d, a program you can try, as a free trial download, at this website.)

View original post

Looking At Things Four-Dimensionally


I’d like to close out the month by pointing to 4D Visualization, a web site set up by … well, I’m not sure the person, but the contact e-mail address is 4d ( at ) eusebeia.dyndns.org for whatever that’s worth. (Worse, I can not remember what site led me to it; if you’re out there, referent, please say so so I can thank you properly. In the meantime, thank you.) The author takes eleven chapters to discuss ways to visualize four-dimensional structures, and does quite a nice job at it. The ways we visualize three-dimensional structures are used heavily for analogies, and the illustrations — static and animated — build what feels like an intuitive bridge to me, at least.

Eusebeia (if I may use that as a name) goes through cross-sections, which are generally simple to render but which tax the imagination to put together1, and projections, and the subtleties in rendering two-dimensional images of three-dimensional projections of four-dimensional structures so that they’re sensible. It’s all quite good and I’m just sorry that my belief in the promise “More chapters coming soon!” clashes with the notice, “Last updated 13 Oct 2008”.

The main page is still being updated regularly, including a Polytope Of The Month feature. A polytope is what people call a polygon or polyhedron if they don’t want their discussion to carry the connotation of being about a two- or three-dimensional figure. It’s kind of the way someone in celestial mechanics talking about the orbit of an object around another might say periapsis and apoapsis, instead of perigee and apogee or perihelion and aphelion, although as far as I can tell people in celestial mechanics are only that precise if they suspect someone pedantic is watching them. I’m not well-versed enough to say how much polytope is used compared to polyhedron.

Anyway, for those looking for the chance to poke around higher dimensions, consider giving this a try; it’s a good read.

[1: I knew that a three-dimensional cube has, on the right slice, a hexagonal cross-section. It’s something I discovered while fiddling around with the problem of charged particles on a conductive-particule sphere, believe it or not. ]