# Rotachora: 4d round and flat shapes

## Last revised 2003-03-16

With the following rotachora intersection images, the left-most object is the face of the 4d shape as it first intersects realmspace (3d space), and the right-most object is the opposite face, as the 4d object is about to leave realmspace. There are two sequences for each 4d shape. The first sequence shows the 4d object passing through realmspace bottom-first, and the second sequence is the shape passing through on its side. The delta (four-dimensional bottom) side of the tetral object is green, and the upsilon (four-dimensional up) side of the tetral object is blue.

### I. The Tetracube

Other names: tesseract, hypercube
Number Series Notation
: 1111
Algebraic Labelling
: (1,4)
Formula
: |x| = 1, |y| = 1, |z| = 1, |w| = 1
Surface Volume: 8a3
Bulk: a4
Vertices
: 16, Edges: 32, Faces: 24, Volumes: 8, Bulks: 1

#### Intersection with realmspace

When a tetracube passes through realmspace cube-face first (as opposed to square-face, edge, or vertex first), the intersection will be a cube. The cube won't change size as the tetracube passes through. The intersection will be the same no matter which cube-face is passed through first.

#### Rolling

The tetracube can't roll anywhere, just like the square and the cube can't roll anywhere.

### II. The Cubinder

Other names: hypercylinder
Number Series Notation
: 112
Algebraic Labelling
: (2,4)
Formula
: x2 + y2 = 1, |z| = 1, |w| = 1
Surface Volume: 2πrh(h + 2r)
Bulk: πr2h2

#### Intersection with realmspace

A cubinder's intersection will behave in different ways depending on how it is inserted into realmspace. When a cubinder is passed through realmspace flat-side first, the intersection is a cylinder. This behavior is like the cylinder passing through planespace flat-side first, whose intersection is a circle. When a cubinder is passed through realmspace round-side first, the intersection will start as a square. It will grow to a cube at the cubinder's midpoint, and then will shrink back down to a square as the cubinder passes all the way through. This is like the cylinder being passed through planespace with its round-side first.

#### Rolling

The cubinder, like the cylinder and the circle, can only cover the space of a line by rolling.

### III. The Duocylinder

Other names: (none)
Number Series Notation
: 22
Algebraic Labelling
: (2,2,4)
Formula
: x2 + y2 = 1, z2 + w2 = 1
Surface Volume: 4π2r3
Bulk: π2r4

#### Intersection with realmspace

A duocylinder's intersection with realmspace is a cylinder. The height of the cylindrical intersection changes as the duocylinder passes through realmspace. First it starts as a circle which grows to a cylinder, then shrinks back down to a circle again. A duocylinder's intersection will be a cylinder no matter which of its sides is passed through first.

#### Rolling

The duocylinder is a peculiar shape. When placed on a surface in tetraspace, a duocylinder can only cover the space of a line by rolling, like a circle, cylinder or cubinder. But, the duocylinder has a round side that is perpendicular to the rolling side in contact with the surface. So, if you turn the duocylinder on its side, it can still roll, but along a different line from the original one. A duocylinder will always be able to roll when placed on a surface in tetraspace, but the space it can cover is only a line.

### IV. The Spherinder

Other names: (none)
Number Series Notation
: 13
Algebraic Labelling
: (3,4)
Formula
: x2 + y2 + z2 = 1, |w| = 1
Surface Volume: 4πr2h + (8/3)πr3
Bulk: (4/3)πr3h

#### Intersection with realmspace

A spherinder's intersection can behave in two different ways, like the cylinder. If the spherinder's flat side is intersected with realmspace, the result is a sphere of unchanging size. This is analogous to the cylinder passing through planespace flat-side first, creating an unchanging circle. If the spherinder is passed through realmspace round side first, the intersection will be a cylinder that grows and shrinks. The duocylinder's intersection is also a growing and shrinking cylinder, but the spherinder's cylindrical intersection behaves differently - the round side of it grows and shrinks instead of the flat side.

#### Rolling

The spherinder is just the sphere extended linearly into tetraspace. This means that it has two dimensions of freedom, and can cover the space of a plane by rolling. The plane that the spherinder can cover can be oriented in different directions, thus changing the shape's rolling space.

### V. The Glome

Other names: hypersphere
Number Series Notation
: 4
Algebraic Labelling
: (4,4)
Formula
: x2 + y2 + z2 + w2 = 1
Surface Volume: 2π2r3
Bulk: (1/2)π2r4

#### Intersection with realmspace

No matter how a glome passes through realmspace, it intersects as a growing and shrinking sphere.

#### Rolling

The glome is analogous to the sphere in realmspace and a circle in planespace - it can roll in any direction that its surface is sloped towards. Thus, it can cover all of a surface in tetraspace by rolling.

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