Question about tiling 4d hyperbolic space with polytopes

[anderscolingustafson]

I understand that in hyperbolic space, in both 2d and 3d it’s possible to tile the plane using shapes that could not on their own tile Euclidean space. For instance in 2d Euclidean space pentagons cannot on their own be used to tile space, but in 2d Hyperbolic space pentagons can be used to tile space all on their own. Similarly in 3d hyperbolic space right angled dodecahedrons can tile the plane all on their own.

I was wondering if in 4d some polytopes that couldn’t tile Euclidean space on their own could tile hyperbolic space on their own. For instance if I understand correctly in 4d Euclidean space the 120 cell could not tile space on its own. Could the 120 cell tile 4d hyperbolic space on it’s own if it had 90 degree angles similar to how 3d dodecahedrons can tile 3d hyperbolic space with 90 degree angles?

[Klitzing]

The 120-cell (or hecatonicosachoron) is being described by the Coxeter-Dynkin diagram x5o3o3o. Its incidence matrix then is

Code: Select all

o3o3o5x

. . . . | 600 |    4 |   6 |   4
--------+-----+------+-----+----
. . . x |   2 | 1200 |   3 |   3
--------+-----+------+-----+----
. . o5x |   5 |    5 | 720 |   2
--------+-----+------+-----+----
. o3o5x |  20 |   30 |  12 | 120


cf. to https://bendwavy.org/klitzing/incmats/hi.htm

For sure that one can be used as the facet of various regular hyperbolic tetracombs. For instance x5o3o3o3o (the (order 3) hecatonicosachoric tetracomb), with incidence matrix

Code: Select all

o3o3o3o5x   (N → ∞)

. . . . . | 120N |    5 |   10 |  10 | 5
----------+------+------+------+-----+--
. . . . x |    2 | 300N |    4 |   6 | 4
----------+------+------+------+-----+--
. . . o5x |    5 |    5 | 240N |   3 | 3
----------+------+------+------+-----+--
. . o3o5x |   20 |   30 |   12 | 60N | 2
----------+------+------+------+-----+--
. o3o3o5x |  600 | 1200 |  720 | 120 | N


cf. https://bendwavy.org/klitzing/incmats/o3o3o3o5x.htm, or for instance x5o3o3o4o (the order 4 hecatonicosachoric tetracomb), with incidence matrix

Code: Select all

o4o3o3o5x   (N → ∞)

. . . . . | 75N |    8 |   24 |   32 | 16
----------+-----+------+------+------+---
. . . . x |   2 | 300N |    6 |   12 |  8
----------+-----+------+------+------+---
. . . o5x |   5 |    5 | 360N |    4 |  4
----------+-----+------+------+------+---
. . o3o5x |  20 |   30 |   12 | 120N |  2
----------+-----+------+------+------+---
. o3o3o5x | 600 | 1200 |  720 |  120 | 2N


cf. https://bendwavy.org/klitzing/incmats/o4o3o3o5x.htm, or even x5o3o3o5o (the order 5 hecatonicosachoric tetracomb), with incidence matrix

Code: Select all

x5o3o3o5o   (N → ∞)

. . . . . |   N |  120 |  720 | 1200 | 600
----------+-----+------+------+------+----
x . . . . |   2 |  60N |   12 |   30 |  20
----------+-----+------+------+------+----
x5o . . . |   5 |    5 | 144N |    5 |   5
----------+-----+------+------+------+----
x5o3o . . |  20 |   30 |   12 |  60N |   2
----------+-----+------+------+------+----
x5o3o3o . | 600 | 1200 |  720 |  120 |   N


cf. https://bendwavy.org/klitzing/incmats/x5o3o3o5o.htm.

--- rk

[quickfur]

One thing to remember about these hyperbolic tilings, that's often neglected to be mentioned, is that they are possible only because we assume arbitrary scaling of the tiles involved, which allows us to choose a scale that matches with the hyperbolic curvature such that the tiles will actually close up into a tiling. The pentagonal tiling of the hyperbolic plane, for example, requires pentagonal tiles of a specific size that's dependent on the curvature of the plane. Only pentagons of this exact size will produce the expected tiling; if you have pentagons of a different size, they will either produce a different tiling (the sizes for which this happens is discrete) or none because they won't close up (the general case).

So when we're talking about tiling hyperbolic 4D space with the 120-cell, we're talking about 120-cells of just the right size that will match the curvature of the space, so that they will close up and form a tiling. The larger the tile, the more tiles you can fit around an edge, due to the curvature of space; so multiple tilings are possible, corresponding with the number of 120-cells you fold around an edge; but each such tiling will require 120-cells of a specific size. In-between sizes will not fit correctly and will not tile space.

In an Euclidean space (of any dimension), scaling a tile does not change the tiling, and if tiles of one scale can produce a tiling, then the same tiles of any scale will produce the same tiling. This is very different from hyperbolic space, where a tileset almost always has multiple tilings, but each one requires a specific tile size and in-between size don't work.