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I am looking for a book that goes deep into the geometry where points have area preferably were points are hexagons but a good introduction of the geometry where points are squares is also welcome.

I would like to learn what he consequences of such an geometry are.

Some I can think of:

A segment contains only a limited number of points.

Segments have an area.

Some parallel lines are indistinguisable (because they are the same point sets)

And I guess many more curious concequences.

I would like te learn about the euclidean case and after that develop my own ideas in the hyperbolic geometry case, but i need a good footing :)

The geometry of hexagons was bij a commenter in an early version of this question refered to as the beehive geometry but a google seach did not give interesting result

I found some pages on hexagonal geometry:

http://hexnet.org/content/hexagonal-geometry

https://en.wikipedia.org/wiki/Hexagonal_tiling

http://hexnet.org/content/hexagonal-geometry

https://en.wikipedia.org/wiki/Hex_map

http://keekerdc.com/2011/03/hexagon-grids-coordinate-systems-and-distance-calculations/

http://www.redblobgames.com/grids/hexagons/

http://hexgridutilities.codeplex.com/

But i would like much more :)

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  • $\begingroup$ So, intuitively, the geometry of a beehive? $\endgroup$ – Arthur Aug 10 '16 at 22:06
  • $\begingroup$ Don't laugh. Loop quantum gravity (en.wikipedia.org/wiki/Loop_quantum_gravity) asserts that what we call points in space are discretized. $\endgroup$ – Oscar Lanzi Aug 13 '16 at 0:03
  • $\begingroup$ @Arthur, i edited my question included your comment thanks $\endgroup$ – Willemien Aug 14 '16 at 10:33

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