Types of 4D shapes

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, I assume we are familiar with various 3D shapes. Some have flat surfaces, and some have curved surfaces. We can categorise these as follows:

• Flat face. Ie. 2D shape.
• Developable curved face. Eg. cylinder, cone.
• Undevelopable curved face. Eg. sphere, torus.

A developable curved surface is curved, yet can be flattened into a 2D shape without stretching. Imagine putting a label on the wine bottle. We can press the paper on the bottle properly without folding it. This means the cylinder face of the wine bottle is developable. Next, let's put a label on the ball for baseball. We can't attach it without distortion. This means the sphere face of that ball is undevelopable.

A 3D shape that is covered only by flat surfaces is called a polyhedron, or multiface body. (Jap: 多面体.) Polyhedra have some properties in common:

• All faces are 2D polygons.
• Have 4 faces or more.
• 2 faces on both sides of any edge. (Called 2-piece.)
• 3 faces or more on any vertex. (Called n-lobed.)

These are useful to calculate the element numbers of shapes. Eg. How many edges and vertices does a cube have? Answer: A cube is made of 6 squares, so 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 an icosahedron have? (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.

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, or multicell body. (Jap: 多胞体.) Polychora have some properties in common:

• All cells are 3D polyhedra.
• Have 5 cells or more.
• 2 cells on both sides of any face.
• 3 cells or more on any edge.
• 4 cells or more on any vertex.

Thinking question: how many faces, edges and vertices does an Ex (600-cell) have? (The Ex 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.

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