horocycle.png copyright 08/18/2002 Wendy Krieger

The horocycle is a most unusual curve. It is a circle of radius infinity, the surface of it has Euclidean geometry. In the poincoire projection that Tyler uses, a horocycle appears as a circle that is tangental to the horizon circle.

You can get a horocycle by using a value of 0, but don't try to insert the {oo} yourself: tyler will take ages to put it in :)

One property is that the content is proportional to surface area. That's right. One can set curvature up so that one foot of horocycle covers an acre, regardless of where it is.

So if your maths is shot, or you don't trust the formulae, you can now count them yourself. What I have presented is four horocyclic honeycombs.

4,3,4,oo is the "infinite apeirogon cupolae".

The point at 8 o'clock, where all the circular layers head off to, is actually a point on the horizon, infinitely far from the action.

• An apeirogon is a polygon with infinity sides.
• A cupolae is a small cup. If you look at two consequative layers, the inner layer has only squares, while the outer layer has alternating squares and triangles. If the inner polygon is a triangle, square or pentagon, the outer polygon will be a hexagon, octagon or decagon, and the result will make a small cup. Since 2 * oo = oo, we can make layer on layer of cupolae.

Just as the 4,3,4,oo produces horocyclic layers that have twice as many in each outer layer, the 6,6,oo and 6,3,6,oo produce layers with three and five times as many members in each outer shell.

The last example shows 8,3,8,oo. This basically follows the same pattern as 6,3,6,oo, but the exact connection happens over at the 9 o'clock point. Because the layers are non-integral, they do not match, either here or anywhere else. This is more the case.

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