As there are already many websites dedicated to 4D geometry, I was originally not gonna write about it. Despite that, I felt like writing for some reason, so I made this article.
First, let's look at various 3D shapes. 3D shapes are covered with 2D shapes, i.e., faces.
In the image above, the cube is covered by only flat faces.
The sphere is covered by only a curved face. The sphere surface is undevelopable, meaning that it can't be flattened without distortion.
The cylinder has both flat faces and a curved face. The curved face is developable, meaning that it's just a rectangle rolled up.
Question: categorize the 3D shapes in the image:
- Which have only flat faces.
- Which have developable curved faces.
- Which have undevelopable curved faces.
A 3D shape that is covered only by flat surfaces is called a polyhedron. (Jap: 多面体 = multiface body.) Polyhedra have some properties in common:
- All faces are 2D polygons.
- Covered by at least 4 faces.
- 2 faces touch any edge.
- At least 3 faces touch any vertex.
The most common & basic 3D shape is the cube, which is covered by 6 squares. 3 faces touch any vertex. With these properties, we can calculate the numbers of elements - edges and vertices - of the cube.
Answer: all faces have 4×6=24 edges and vertices in total. 2 of these edges form an edge of a cube. Thus, a cube has 24/2=12 edges. 3 faces meet at a vertex of a cube. Thus, a cube has 24/3=8 vertices.
Thinking question: how many edges and vertices does the icosahedron have? Hint: the icosahedron is made by 20 triangles; 5 faces meet at any vertex.
Polyhedra with the fewest faces are triangular pyramids, which are made of 4 triangles.
In 3D geometry, terms like regular, uniform, and noble, are always applied to polyhedra. Thus, I simply call them just 3D shapes when there's no confusion.
Various 4D shapes
4D geometry is the natural extension of 3D geometry. We explore the diversity of 4D shapes.
We have to imagine the fourth dimension, which is just another direction extended outside our 3D space. 4D Euclidean space is challenging to imagine for most people. In history, people are fascinated by the imaginary 4D world. Today, definitely there are people who can see the 4D.
4D shapes are covered by 3D shapes, just like 3D shapes are covered by 2D shapes. 3D elements of 4D shapes are usually called cells. (Jap: 胞.) As in 3D, we can classify cells as this:
- Flat cell. Ie. 3D shape.
- Developable curved cell. Eg. hypercylinder, hypercone.
- Undevelopable curved cell. Eg. hypersphere.
A "curved cell" is a 3D shape curved in 4D space. It has the same properties as a flat 3D shape, but we already can't imagine it clearly.
A 4D shape that is covered only by flat "surcells" is called polychoron. (Jap: 多胞体 = multicell body.) Polychora have some properties in common:
- All faces are 3D polyhedra.
- Covered by at least 5 cells.
- 2 cells touch any face.
- At least 3 cells touch any edge.
- At least 4 cells touch any vertex.
Thinking question: how many faces, edges and vertices does the hexacosichoron (600-cell) have? (The hexacosichoron is made by 600 regular tetrahedra; 5 cells meet at any edge; 20 cells meet at any vertex.)
Polychora with the fewest cells are "tetrahedral pyramids," which are made by 5 triangular pyramids.
In 4D geometry, terms like regular, uniform, and noble, are always applied to polychora. Thus, I simply call them just 4D shapes when there's no confusion.