Welcome back to Numberholic! I’m your host, Sejari, and it’s always a pleasure to have you here. Today, we’re diving into the tenth episode of Proven Yet Hard to Believe, a series where we uncover the fascinating stories and directly explore the core ideas behind mathematical discoveries that have stood the test of time.
If there are regular polyhedra in three dimensions, then in four dimensions, there are regular polytopes. Let’s explore what four-dimensional regular polytopes are and what they look like.
A two-dimensional polygon consists of points and lines, with lines as its boundaries. It encloses a single planar region.
A three-dimensional solid consists of points, lines, and faces, with faces as its boundaries. It encloses a spatial region.
Thus, a four-dimensional polytope must have points, lines, faces, and solids, with solids as its boundaries. It encloses a four-dimensional region.
A four-dimensional regular polytope is a higher-dimensional counterpart of a three-dimensional regular polyhedron. All the solids that form its boundaries are regular polyhedra.
Just as there are only five types of regular polyhedra in three dimensions, the number of four-dimensional regular polytopes is also limited—there are exactly six.
The number in the name of a regular polyhedron refers to the number of faces, while the number in the name of a regular polytope refers to the number of three-dimensional solids forming its boundaries.
For example, the 5-cell (or pentachoron) is a four-dimensional polytope where all five boundary solids are regular tetrahedra.
Each regular polytope can be derived from a three-dimensional regular polyhedron:
- The 5-cell (pentachoron) comes from the tetrahedron.
- The 8-cell (tesseract) comes from the cube.
- The 16-cell comes from the octahedron.
- The 120-cell comes from the dodecahedron.
- The 600-cell comes from the icosahedron.
However, the polyhedron from which a polytope is derived is not necessarily the same as the polyhedra forming its boundaries.
For example, the 16-cell, which is derived from the octahedron, has tetrahedral boundaries.
But there is something even more unique: the 24-cell.
This polytope has no counterpart in three dimensions, nor in five dimensions—it exists solely in four dimensions.
Because of this, it is often referred to as “the unique four-dimensional regular polytope.”
Now, let’s construct this polytope.
The 24-cell consists of octahedral boundaries.
Let’s start with an octahedron.
Do you remember how we constructed regular polyhedra by filling 2-dimensional space seamlessly?
To form a 24-cell, we attach three octahedra to each edge, filling 3-dimensional space as we go.
We attach one octahedron to each face, ensuring they fit perfectly together.
In total, we can attach eight.
This leaves six gaps.
Now, let’s ignore the gaps for a moment and attach another eight octahedra.
When attaching new octahedra to the original octahedron’s faces, you can see that the triangles are rotated by 180 degrees.
Attaching another set of octahedra results in another 180-degree rotation, forming a large octahedral shape.
Since we are visualizing a four-dimensional situation in three dimensions, the octahedra appear inflated and enlarged.
But two key things are clear:
- Attaching eight more octahedra creates a larger octahedral shape.
- The previously ignored gaps are also filled with octahedra—though their proportions may not appear perfectly accurate. We insert one octahedron into each of the six remaining gaps.
By assembling 1+8+8+6 octahedra, we construct a large octahedral structure.
Now, using a single inverted octahedron, we complete the three-dimensional space.
And with that, we are essentially done.
ㅉith 24 octahedra in total, we have completely filled the space.
Ah, and don’t forget the final dot.
That’s all for today’s episode of Proven Yet Hard to Believe! If you enjoyed this deep dive, don’t forget to subscribe and stay tuned for our next journey into the world of mathematics. Until next time, keep exploring, keep questioning, and stay Numberholic!