# Rotahedra: 3d round and flat shapes

## Last revised 2003-03-16

### I. The Cube

Number Series Notation: 111
Algebraic Labelling
: (1,3)
Formula
: |x| = 1, |y| = 1, |z| = 1
Surface Area: 6a2
Volume: a3
Vertices
: 8, Edges: 12, Faces: 6, Volumes: 1

#### Intersection with planespace

When a cube intersects with planespace, a square is created. The square doesn't change size as the cube passes through.

#### Rolling

A cube, like the square, can't roll anywhere when placed on a surface.

### II. The Cylinder

Number Series Notation: 12
Algebraic Labelling
: (2,3)
Formula
: x2 + y2 = 1, |z| = 1
Surface Area: 2πrh + 2πr2
Volume: πr2h

#### Intersection with planespace

A cylinder's intersection with planespace behaves in different ways depending on the cylinder's orientation with respect to planespace. If a cylinder passes through planespace on its side, it will appear to be a rectangle that grows then shrinks. This is like a circle passing through linespace. If you instead insert the cylinder flat side first, it will appear to be a circle that doesn't change in size as the cylinder passes through. This is like a square passing through linespace.

#### Rolling

The cylinder, like the circle, can only cover the space of a line by rolling. But, the cylinder can be oriented in different directions, thus changing where the shape can roll to. By using only rolling, the cylinder cannot cover all of a surface in realmspace.

### III. The Sphere

Number Series Notation: 3
Algebraic Labelling
: (3,3)
Formula
: x2 + y2 + z2 = 1
Surface Area: 4πr2
Volume: (4/3)πr3

#### Intersection with planespace

A sphere will look the same when intersected with planespace no matter what its orientation. As it is intersected, it will first appear as a point that grows to a circle. After the sphere has passed halfway through planespace, the circle shrinks back down to a point.

#### Rolling

A sphere can roll in any direction that its surface is sloped towards. Thus, it can cover all of a surface in realmspace by rolling. This is similar to how a circle can cover all of a surface in planespace by rolling. But, the sphere's space of coverage is a plane instead of a line like the cylinder or circle.

The rendered images on this page were created by rt, an n-dimensional raytracer, from nklein software.