# Number of lines and 2 dimensional sides on a tesseract **URL:** \ **Category:** Factual Questions\ **Created:** [May 11, 2021, 9:02pm UTC](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868 "2021-05-11T21:02:38Z")\ **Posts on this page:** 6\ **Page:** 2
**Author:** ![mikecurtis](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mikecurtis/32/3284_2.png) [@mikecurtis](https://boards.straightdope.com/u/mikecurtis)\ **Post date:** [May 13, 2021, 2:28am UTC](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868/21 "2021-05-13T02:28:44Z")
I’ve found this video very helpful in visualizing higher dimension solids (polytopes) [![](https://img.youtube.com/vi/2s4TqVAbfz4/maxresdefault.jpg "Perfect Shapes in Higher Dimensions - Numberphile") ](https://www.youtube.com/watch?v=2s4TqVAbfz4) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 13, 2021, 3:06am UTC](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868/22 "2021-05-13T03:06:53Z")
For super bonus extra fun, you can try using complex numbers instead of real numbers. Things are already pretty cray in 2 (complex!) dimensions [therefore something like 4 real dimensions] because a _complex_ edge can intersect more than 2 vertices. Now, instead of a tesseract, you have a configuration with 16 vertices and 8 4-edges. But there are many new possibilities… ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [May 13, 2021, 7:12am UTC](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868/23 "2021-05-13T07:12:43Z")
Once discussion of the physical form of hyper cubes comes up I can’t help but cite the CM-1 and CM-2. [https://tamikothiel.com/theory/cm\_txts/index.html](https://tamikothiel.com/theory/cm_txts/index.html) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 13, 2021, 8:42am UTC](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868/24 "2021-05-13T08:42:56Z")
I remember in the 1980s IBM was experimenting with SIMD supercomputers with however many processors interconnected by a dynamically reconfigurable Clos network, so that at each machine cycle the connections can be set to a different permutation (so you can use a rectangular mesh of any dimension, hexagonal mesh, torus, whatever). The switches in this type of network are themselves recursively arranged in an interesting way: [![](https://upload.wikimedia.org/wikipedia/commons/c/c9/Benesnetwork.png) ](https://upload.wikimedia.org/wikipedia/commons/c/c9/Benesnetwork.png) ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [May 13, 2021, 10:23pm UTC](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868/25 "2021-05-13T22:23:55Z")
I will not give the proof of this, but @senegoid’s post above provides the explanation. The number of k-dimensional “faces” (that’s the technical term) in an n-dimensional cube is the coefficient of x^ky^(n-k) in the binomial expansion of (x + 2y)^n. Since (x+2y)^5 = x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^5 + 32y^5, a 5-cube has 1 five dimensional face (itself), 10 4-dimensional faces (tessaracts) 40 3D faces (ordinary cubes), 80 2D faces (squares), 80 edges, and 32 vertices. Of course, the y is irrelevant; you could use (x + 2)^n instead. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [May 14, 2021, 2:22pm UTC](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868/26 "2021-05-14T14:22:07Z")
At the risk of over-explaining this, replace (x+2y)^n above by (e+2v)^n. e stands for edge and v for vertex. An n-dimensional cube is, in a very precise sense, a product of n line segments and a line segment has one edge and 2 vertices. A k-dimensional face arises when k of the factors are edges and n-k of them are vertices and the coefficients of e^kv^(n-k) tells how many ways to do that. [Previous page](https://boards.straightdope.com/t/number-of-lines-and-2-dimensional-sides-on-a-tesseract/941868.md?page=1)