# Rotahedra: 3d round and flat shapes

« Site Map « Shapes of the dimensions « Rotatopes

## 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**: 6a^{2}

**Volume**: a^{3}

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: x^{2} + y^{2} = 1, |z| = 1

**Surface Area**: 2πrh + 2πr^{2}

**Volume**: πr^{2}h

#### 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: x^{2} + y^{2} + z^{2} = 1

**Surface Area**: 4πr^{2}

**Volume**: (4/3)πr^{3}

#### 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.

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