*Below, a square is a regular 4-gon. In other words,
it is a 4-gon with all
the sides and all the angles congruent. (For simplicity, we assume
that the geometry is over real numbers. This excludes geometries of positive
curvature, which require non-Archimedian base fields!).*

In the Lobachevsky flavors of non-Euclidean geometries, one can tune the size of the square so that the angles become any given angle below 90°. In particular, it can be exactly 72°. Then one can fit FIVE of such corners together so that they total the full angle; this joins 5 squares into a star-like shape. There are corners of this shape where two squares meet; adding 3 more squares to these corners, one gets a full angle filled at these corners too. One can continue this process indefinitely; as a result, the Lobachevski plane is covered by a “square grid” — with 5 squares meeting at every corner.

*BTW, one can replace 5 above by 6,7, etc too.*

I could find only a “cheating”
existing picture of such a square grid. The problem with this image is a general
problem with human-friendly visualizations of Lobachevsky geometry: in our
world, on human scale the geometry of the universe **is** very close to
a flat geometry. (One needs to go to the scale of tens of millions of light year
to see non-flatness — unless one cheats and uses time-like directions too; then
the curvature is visible already on the scale of light-minutes: if you throw
a used soda can out of a window of a moving satellite, it may return back after
89 minutes! The same number appears as instability of inertial navigation systems,
and both effects are due to our geometry being “non-flat”. **To add insult to confusion:** one effect is due to “positive” curvature, another to “negative” curvature!)

Anyway, to draw human-scale visualizations of Lobachevsky geometry, one must use certain kinds of “distortions”. (This is a complete analogue of why one cannot draw “a honest map” of a part of a globe; the analogy may be substantiated by the “Wick rotation”.)

In the “cheating” picture mentioned above, to visualize straight
lines on Lobachevsky plane, they use “usual” straight lines. While it is
possible to do this self-consistently (“Klein
model” of Lobachevsky plane), in such “a honest” visualization the
squares are drawn extremely “thin” if we go just a few steps aways from the origin.
In *our* Klein-model picture,
one can zoom to see what happens away from
the center, the “thinness” is so overwhelming that going more than 6 steps from the
boundary, the images of squares are hidden by the thickness of lines. Using
thiner lines
helps (but with thiner lines many PDF renderers show the PDF with some lines
missing! However, `Acrobat`

I checked can show it fine).

The “cheating” picture mentioned above does not use the Klein model, so it does not
give any “photo-realistic” rendering of how the squares are positioned w.r.t. each
other. For examples, straight lines which should be continuations of each other
are not in this rendering. (But thanks to abandoning “photo-realism”, they manage
to draw *a lot* of squares this way!)

OK, since we could not find a readily available “honest” picture, we designed software to draw such pictures. This way, we got the pictures in Klein model linked to above, and the pictures in “the other”, Poincaré model.

The “Poincaré model” is easier to the eye, but the straight lines in Lobachevsky geometry are drawn curved in this flavor of visualization. However, it is still “photo-realistic” in the sense that the angles are visualized correctly. In particular, our 5-grid in Poincaré model has every quadrilateral with 4 angles of 72° each.

While one can see many more squares in this visualization, I must say that it still in a “non helpful” category. I hoped to get some suggestive pictures so that parents of kids in grades 3—4 could have a chance (maybe a remote chance, but still a chance) to invent something to discuss with their kids when we cover the result “sum of angles of △ is 180° — as far as the triangle is drawn on a square grid”. Currently, I do not see how one could use these pictures for these purposes. (If you can, please let me know! Probably by using all 3 flavors together: Klein/Poincaré/cheating?)

**P.S.** While not helpful for Math Circles Elementary, these pictures
gained an unexpected benefit: serendipiously, many points on the Poincaré model
picture align along straight lines! This has a certain chance to be a “new discovery”!
See the same image with red guide lines, and points which are on the guide lines
marked by minuscule blue circles. (We check that a point is on a guide line
with precision of 100 decimal places.)