Platonic Solids and the Quaternions

While you're pondering that, let me tell you another way to get some of the 4d regular polytopes. This method involves quaternions, which are a souped-up version of the complex numbers with three square roots of -1, called i, j, and k. A typical quaternion looks like this:

a + bi + cj + dk

where a,b,c, and d are real numbers. To multiply the quaternions, you need to use these rules, invented by Hamilton back in 1843:

i2 = j2 = k2 = -1



ij = -ji = k

jk = -kj = i

ki = -ik = j

Let's start with the 24-cell, since this guy has no analog in other dimensions. Since the vertices of the 24-cell lie on the unit sphere in 4 dimensions, we can think of its vertices as certain unit quaternions. The 24-cell happens to have, not only 24 faces, but also 24 vertices! We can take them to be precisely the unit 'Hurwitz integral quaternions', which are quaternions of the form

a + bi + cj + dk

where a,b,c,d are either all integers or all integers plus 1/2. One can check that the Hurwitz integral quaternions are closed under multiplication, so the vertices of the 24-cell form a subgroup of the unit quaternions. A regular polytope that's a symmetry group in its own right - ponder that while you cross your eyes and gaze at it spinning around!

Similarly, the 600-cell has 120 vertices, which we can think of as certain unit quaternions. We can take them to be precisely the unit 'icosians'. These are quaternions of the form

a + bi + cj + dk

where a,b,c,d all live in the 'golden field' - meaning that they're of the form x + √5 y where x and y are rational. Since the icosians are closed under multiplication a group under multiplication, the vertices of the 120-cell also form a group!

The vertices of the 4-dimensional cross-polytope also form a subgroup of the unit quaternions. But this one is a little less exciting. We just take the quaternions of the form

a + bi + cj + dk

where one of the numbers a,b,c,d is 1 or -1, and the rest are zero. This 8-element subgroup is sometimes called 'the quaternion group'.

Those are all the 4-dimensional regular polytopes that are also groups. Three out of six ain't bad! But we can get most of the rest using duality.

In general, the 'dual' of a regular polytope is another polytope, also regular, having one vertex in the center of each face of the polytope we started with. The dual of the dual of a regular polytope is the one we started with (only smaller). So polytopes come in mated pairs - except for some 'self-dual' ones.

In 2 dimensions, every regular polytope is its own dual.

In 3 dimensions, the tetrahedron is self-dual. The dual of the cube is the octahedron. And the dual of the dodecahedron is the icosahedron.

In 4 dimensions, the 4-simplex is self-dual. The 24-cell is also self-dual - that's why it had 24 faces and also 24 vertices! The dual of the hypercube is the 4-dimensional cross-polytope. The dual of the 120-cell is the 600-cell.

In higher dimensions, the n-simplex is self-dual, and the dual of the n-cube is the n-dimensional cross-polytope.

But what is so special about 4 dimensions, exactly?

Well, there are very few dimensions in which the unit sphere is also a group. It happens only in dimensions 1, 2, and 4! In 1 dimensions the unit sphere is just two points, which we can think of as the unit real numbers, -1 and 1. In 2 dimensions we can think of the unit sphere as the unit complex numbers, exp(iθ). In 4 dimensions we can think of the unit sphere as the unit quaternions.

Only in these dimensions do we get polytopes that are also groups in a natural way. In 2 dimensions all the regular n-gons correspond to groups consisting of the unit complex numbers exp(2πi/n). In 4 dimensions things are more subtle and interesting. It's especially interesting because the group of unit quaternions, also known as SU(2), happens to be the 'double cover' of the rotation group in 3 dimensions. Roughly speaking, this means that there is a nice function sending 2 elements of SU(2) to each rotation in 3 dimensions.

This gives a slick way to construct the 600-cell, or hypericosahedron. Take the icosahedron in 3 dimensions. Consider its group of rotational symmetries. This is a 60-element subgroup of the rotation group in 3 dimensions. Now look at the corresponding subgroup of SU(2) - its 'double cover', so to speak. This is a 120-element subgroup of the unit quaternions. These are the vertices of the hypericosahedron! So in a very real sense, the hypericosahedron is just the symmetries of the icosahedron! This trick doesn't work in higher dimensions. This is one thing that's very cool about 4 dimensions - it inherits the hypericosahedron and the hyperdodecahedron from the the fact that the icosahedron and dodecahedron happen to exist in 3 dimensions.

Similarly, the 24-cell comes from the symmetries of the tetrahedron!

Directions for Further Study I copied the pictures of rotating Platonic solids from the Wikipedia article on Platonic solids under the terms of the GNU Free Documentation License: these were made by Cyp. I also copied the pictures of 4d regular polytopes from the Wikipedia articles on these polytopes, under the terms of the relevant copyrights: these were made by Tom Run using Robert Webb's Stella software. These articles are a great place to get started on understanding the Platonic solids. For more information try Eric Weisstein's Mathworld website. He has lots of information on Platonic solids and 4d geometry. You can rotate the Platonic solids and 4d polytopes using your mouse! If you have access to VRML, you can also have fun with George Hart's Encyclopedia of Polyhedra, which has over 1000 polyhedra in it. (VRML stands for "virtual reality modelling language", and it's available as a plugin for most browsers.) If you want to learn a lot about regular polytopes, read this book by the king of geometry: H. S. M. Coxeter, Regular Polytopes, 3rd edition, New York, Dover Publications, 1973. For more on the dodecahedron, see "Tales of the dodecahedron: from Pythagoras through Plato to Poincaré".

For more about the icosahedron, see "Some thoughts on the number six".

For more about icosians and related marvels, see week20 of This Week's Finds.

For more about the Platonic solids, how fool's gold fooled the Greeks into inventing the regular dodecahedron, and highly symmetric structures in higher dimensions, see week62, week63, week64, and week65 of my weekly column on mathematical physics. This story continues at a deeper level in week186 and week187.

For more on the Hurwitz integral quaternions and the mysteries of triality in 8 dimensions, see week91.

For a deeper look at relations between different Platonic solids, and also more stuff about the 24-cell and 600-cell, see week155.

Everything sufficiently beautiful is connected to all other beautiful things! Follow the beauty and you will learn all the coolest stuff. The Platonic solids are a nice place to start.



From Kepler's

in which he modeled the orbits of the five known planets using Platonic solids.

The cube fits outside all those shown above. From Kepler's Mysterium Cosmographicium in which he modeled the orbits of the five known planets using Platonic solids.The cube fits outside all those shown above.

© 2020 John Baez

baez@math.removethis.ucr.andthis.edu

