Cubinder

In four-dimensional geometry, the cubinder (otherwise known as cubical cylinder or hypercylinder) is one way to generalise the 3D cylinder to 4D. Like the duocylinder and spherinder, it is also analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment.

${\displaystyle D=\{(x,y,z,w)\mid x^{2}+y^{2}=1,|z|=1,|w|=1\}}$[1]

Construction of a Cubinder by extrusion of a cylinder along the W axis by a unit distance.

The Cubinder is the Cartesian Product of a circle and a square, and it is a rotachoron. It can be constructed by extrusion of a 3D cylinder along the W- axis by a unit distance.[2] The net of a cubinder is given by a single cube, surrounded by 4 cylinders.

Geometry

Formulae

${\displaystyle {\text{volume}}=2\pi rh(h+2r)}$

${\displaystyle {\text{bulk}}=\pi r^{2}h^{2}}$[3]

Subfacets

Tera (4D)	1 cubinder
Cells (3D)	4 cylinders and 1 cell bounded by a square torus
Faces (2D)	4 discs and 4 tubes
Edges (1D)	4 circles
Vertices (0D)	none

Rolling

The square torus that binds the cubinder forms a circular surface on which the cubinder can roll. Like a circle and a cylinder, it can only roll the space of a line. The cubinder cannot roll on the 4 cylinders, as they are flat in 4D.[4]

One perspective projection of the cubinder.

Projection

The diagram is a perspective projection of the cubinder. The cubinder is rotated 45 degrees in the ZW plane. This allows us to observe that the cubinder is made up of four cylinders. However the square torus joining the cylinders cannot be observed from this perspective.

Relationship to other shapes

In 4-space, there are three intermediate forms between the tesseract (1 ball × 1 ball × 1 ball × 1 ball) and the hypersphere (4-ball). These are as follows:

• cubinder (2-ball × 1-ball × 1-ball)
• spherinder (3-ball × 1-ball)
• duocylinder (2-ball × 2-ball).

All five correspond to the integer partitions of the number of dimensions (4); it is easy to see that this also holds for lower dimensions.

The cubinder is also a rotatope (more specifically a rotachoron), along with other shapes such as the tesseract, duocylinder, spherinder, and glome. A rotachoron is a four dimensional shape which can by formed by extensions and rotations.[5] The cubinder is a rotachoron as it can be formed via extension of a cylinder or rotation of a cube.

Cubinders relation to other rotatopes[6]

Dimension	n-type	n-cube	m-sphere crosses	n-sphere	n-ball
2nd	rotagon	square		circle	disk
3rd	rotahedron	cube	cylinder	sphere	ball
4th	rotachoron	tesseract	cubinder, duocylinder, spherinder	glome	gongyl

The rotation of cubinder to 5D is non-unique: rotating it around a cubic cross-section will lead to the shape, spherisquare. However, if you rotate it around its cylindrical cross-section, you will get a different rotatope called "dual cylinder", which can be also gotten by extending the duocylinder.[7]

See also

• Four-dimensional space
• Cartesian product

References

1. Chandra, Satapathy Suresh; Biswal, Bhabendra Narayan; Udgata, Siba K.; Mandal, J. K. (2014-10-31). Proceedings of the 3rd International Conference on Frontiers of Intelligent Computing: Theory and Applications (FICTA) 2014. Springer. ISBN 9783319120126.
2. Banerjee, Agnijo. "Four dimensional space". www.agnijomaths.com. Retrieved 2017-01-19.
3. "Rotachora". hi.gher.space. Retrieved 2017-01-21.
4. "The Cubical Cylinder". eusebeia.dyndns.org. Retrieved 2017-01-19.
5. "Appendix:List of protologisms/Q–Z – Wiktionary". en.wiktionary.org. Retrieved 2017-01-21.
6. "Fourth Dimension Glossary". hi.gher.space. Retrieved 2017-01-21.
7. "Hi.gher. Space • View topic – Disagreement with the Duocylinder". hi.gher.space. Retrieved 2017-01-21.