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]
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.
Construction of a Cubinder by extrusion of a cylinder along the W axis by a unit distance.
Geometry
Formulae
volume = \( {\displaystyle {\text{volume}}=2\pi rh(h+2r)} \)
bulk = \( {\displaystyle {\text{bulk}}=\pi r^{2}h^{2}}[3] \)
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.
| 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 |
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
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.
Banerjee, Agnijo. "Four dimensional space". www.agnijomaths.com. Retrieved 2017-01-19.
"Rotachora". hi.gher.space. Retrieved 2017-01-21.
"The Cubical Cylinder". eusebeia.dyndns.org. Retrieved 2017-01-19.
"Appendix:List of protologisms/Q–Z – Wiktionary". en.wiktionary.org. Retrieved 2017-01-21.
"Fourth Dimension Glossary". hi.gher.space. Retrieved 2017-01-21.
"Hi.gher. Space • View topic – Disagreement with the Duocylinder". hi.gher.space. Retrieved 2017-01-21.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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