# Sphere hexagon subdivision

Hi,

Is there any grasshopper add-on that could help to have a hexagonal subdivision on a sphere.
The image below has 5-side polygon at singularities. I tried smoothing isosahedron from weaverbird, but it does create smooth sphere.

I could be wrong but is it possible that it is impossible to have just hexagon on a sphere.
https://www.mathsisfun.com/geometry/eulers-formula.html
For many solid shapes the

• Number of Faces
• plus the Number of Vertices
• minus the Number of Edges

always equals 2

n +v -e = 2
e = n/2 (each edges touches 2 faces)
v = n/3 (points touches 3 faces)
n+n/3-n/2 = 2
=> n*(6+2-3) =2*6
=> n *5 = 12 thatâ€™s impossible for integer

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You beat me in speed!
You are definitively not wrong.

A solution could be to â€śpackâ€ť all the singularities on a small area of the sphere, leaving the rest with only hexagonsâ€¦ but it would be un-evenly sized hexagons, with the largest on the opposite side of the singularities.

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Thereâ€™s a nice interactive model of a hexagonal tiling of the sphere here:
http://pub.ist.ac.at/~edels/hexasphere/
(note the date of publication)

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Also - if you donâ€™t require the vertices to all be valence 3, and allow adjacent faces to share more than one edge, you can divide a sphere into all 6 sided patches like this:

Just out of curiosity, I tried with RemeshByColor with a mesh with square sine gray scaleâ€¦ but it completely didnâ€™t work. Even some attempt of sphere packingâ€¦ ugly resultsâ€¦

So i switched to pure math:

( I also found interesting this: https://math.stackexchange.com/questions/2121175/is-it-possible-to-have-a-spherical-object-with-only-hexagonal-faces )

And soâ€¦

Stereographic_projection.gh (27.4 KB)

As expected, with the camera at the correct position, the hexagons look completely â€śflatâ€ť:

Interesting and unexpected (for me) thing: the hexagons are ALWAYS flat!
In the definition there is the control to chose where to put the â€śfirstâ€ť hexagon: centered, mid-edge or vertexâ€¦ but the faces are always flat regardless of the starting position!

If needed, it should be doable to â€śpatchâ€ť the hole with Delanuay on the dual of the hexagons, then pulling the result to base surface, joining the meshes and lastly doing the dual againâ€¦

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Thereâ€™s a component for this in Kangaroo now:

The hexagons are always flat because MĂ¶bius transformations (which inversions are) preserve circles. Since the initial regular hexagons have all their vertices lying on a circle, so do the transformed ones, therefore they are planar.

Also - hereâ€™s another covering of the sphere with only 6 sided patches -
fold a hexagon in half along a line joining a pair of opposite vertices, glue the sides of the boundary to each other, and inflate it:

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Wow!

â€śhow to discover a component: the stupid wayâ€ť

Well, itâ€™s always good to find and learn new stuff.

Thatâ€™s my fault - I quietly added this one in the last release, but hadnâ€™t properly documented it yet, except through posts like this.

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I tried itâ€¦ itâ€™s great, nothing to say.
But the sphere radius is hard to grasp.
Having the possibility to use a sphere surface directly would be great. (or center + radius)
Inversions â€śinâ€ť a circle or â€śinâ€ť a sphere are easier to understand.
A circle + T + Q + F parameters is â€śalienâ€ť , to me at least â€¦

You can ignore Q and F for these purposes.

For inversions, the T angle parameter needs to be set as Ď€
This results in an inversion in the sphere which has the circle as its equator (which is also a 2d inversion in the circle in that plane)

For stereographic projection from the 2-sphere to the plane, you can use the equator of the sphere as the circle and set the T angle as Ď€/2

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Nice! â€¦ thank youâ€¦
(I had to set F=true though)

I have 2.42 version of kangaroo, where can I download version with this component included. Food4rhino has latest 2.42 version only.

Itâ€™s only in the rhino 6 version for now. If you have a current R6 service release you should already have it

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I have rhino 6 SR 5 version, so I need to upgrade I think

The 2.5 release is actually on f4r too - right at the bottom of the downloads list, but simpler option is just to get the SR

Thank you.