## 5D is where everything happens?

Higher-dimensional geometry (previously "Polyshapes").

### 5D is where everything happens?

A lot of things seem to happen in 5D:

1) There's only 3 regular polytopes in 5D and above

2) there infinitly many Johnson polytopes in 5D and above (cartesian product of polygons to the Johnson solids and above)

3) the demicube ceases to be regular in 5D and is the only dimension where it is semiregular (regular faceted)

4)the unit-radius hypersphere is at maximum volume in a dmension between 5 and 6 but in whole numbers 5 is the largest size.

5) uncomfirmed: the sausage conjecture states that in 5D and above the comvex hull of any number of hypersheres is smallest when they're arranged in a line (i.e. when the convex hull is a sausage)

what else?
wintersolstice
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### Re: 5D is where everything happens?

wintersolstice wrote:[...]5) uncomfirmed: the sausage conjecture states that in 5D and above the comvex hull of any number of hypersheres is smallest when they're arranged in a line (i.e. when the convex hull is a sausage)

what else?

This is an old topic, but I just noticed something funny today. The sausage conjecture is currently only proven for dimensions n≥42, even though the conjecture itself is stated for n≥5. In Douglas Adam's "Hitchhiker's Guide to the Galaxy", the answer to "life, the universe, and everything" is (humorously) computed to be 42. Upon which, the baffled protagonists embark on a quest to find out what the question was (since "life, the universe, and everything" isn't really a question per se). Now, thanks to this forum, we know what the question was: it was "What's the smallest dimension for which the Sausage Conjecture was proven to be true?"
quickfur
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### Re: 5D is where everything happens?

What is a demicube?
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions

Prashantkrishnan
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### Re: 5D is where everything happens?

Prashantkrishnan wrote:What is a demicube?

If you imagine a cube as convex hull of all 2^n points with each coordinate either +1 or -1, demicube would be a convex hull of half of those points -- only those with, say, even number of +1's in their coordinates.
The difficulty with the demicube is that in 3D and 4D they are actually regular polytopes -- tetrahedron in 3D and 16-cell in 4D. However, in 5D and higher, demicube is its own special uniform shape. The 5D demicube is built from pentachora and 16-cells. Generally, n-dimensional demicube is built from (n-1)-dimensional simplexes and demicubes.
Marek14
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