Not Being Square

I meant to post today a sequel to the circular triangles from last week, but I got carried away looking at right angled quadrilaterals bounded by circular arcs. Like the pillows, but more general. Like so:

Circulon 2

The question arises for what choices of four points we can find a right angled quadrilaterals bounded by circular arcs?

By the way, how do we call these? I thought about circulons (taken) and horny squares (oops). For now, I call them circulions (like centurions), to avoid a lawsuit about trade marks. Above you see a solution that is not a square but where the vertices are at the corners of a square. There are more like these, in fact a 1-parameter family.

Circulon 1

Below you can see the entire family at once, you just have to follow all dots with the same color.


Can we do that for any choice of four points? Not so, but: Möbius allows us to move three points anywhere we like (and he will send circulions to circulions), so we can ask: where are we allowed to place a fourth point so that there is a circulion through all of them?

Möbius also tells us that this is easy if we place all four points on a circle (by sending that circle to a line, and then connecting the four points on the line alternatingly by segments of the real line and half circles, for instance). Here is an example where the first three points are at the corners of an equilateral triangle, and the fourth point is on the circle through them.

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Again, there is a 1-parameter family of such circulions through these points.


Pretty, isn’t it?

Now, surprisingly to me, for each choice of three points, there is a second circle on which the fourth point can reside: Take the circle that contains the given three points, and construct the circle orthogonal through it that passes through the two points between which we want to put the fourth vertex. You can put the fourth point anywhere on that new circle. Here is an example, with the first three points again at the corners of an equilateral triangle.

Circulon 4

Below is again an entire family, color coded and adorned with moiré.


Next week you’ll see conformally correct squares. Promised.


Resources? (Cave River Valley II)

The main attraction of the Cave River Valley Natural Area are not so much its signs of abondonment, but rather its caves and rock formations.

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The area was acquired at some point by the Nature Conservancy, and then possession was transferred to the Department of Natural Resources of Indiana, who had created a site management plan that is an interesting document in many respects.

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It explains in detail what the DNR planned to do with the area, and what the costs would be. The plan did not move forward much, be it because of budget problems, be it because of myotis sodalis, the endangered Indiana Bat.

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The bat uses the Endless Cave above and below as a hibernaculum (I need one, too!). Plans to take busloads of spelunkers through the extensive caves in the area would possibly run afoul of the Federal Cave Resources Protection Act from 1988.

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So the Department of Natural Resources put up a handful of signs and dumped truckloads of gravel on a pathway that was supposed to give access to campsites for up to 120 people. Hmmm.

Then, they abandoned the site, once again.

It is, however, as I hope the pictures are hinting, of some beauty.

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Safely Footed Spiderwebs

… et je continue encore de fouler le parvis sacré de votre temple solennel…

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Let’s talk again a little about triangles. The last time I wrote about triangles is not quite a year ago, and it didn’t help.

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What you see in this post are all triangle that have their vertices at the same place, the third roots of unity, to be precise. They are, however, not Euclidean triangles with straight edges, but curved ones, with circular edges.

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The first three are equiangular still, making angles of 10, 90 and 240 degrees at the corners, respectively. The spiderwebs are conformal images of polar coordinates on the disk, thus illustrating the Schwarz-Christoffel formula for circular polygons. The bat down below is a neat optical illusion, too: Would you think that the vertices are at the corners of an equilateral triangle?

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The theory behind this is based on Schwarzian derivatives and the Schwarz reflection principle, so clearly Hermann Amandus Schwarz owns all this.
It is also intimately connected to hypergeometric functions and much more recent mathematics.

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And there is some mystery, still. While circular triangles are safe (they are determined by their angles, up to a Möbius transformation, and the Schwarz-Christoffel formula will deliver), quadrilaterals are not. Even Euclidean straight & right quadrilaterals can be differently shaped rectangles, and things get worse with circular ones. In this case, the Schwarz-Christoffel formula will have some extra parameters, the so-called accessory parameters. Changing the will not change the conformal nature of the quadrilateral nor its angles, but its “bulge”. More about this later.

Abandonment (Cave River Valley I)

The hilly and not so fertile landscape of southern Indiana offered the early settlers enough room to get by, after the natives had been – what is the euphemism these days – deported?

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With a bit of luck you could find yourself a stream in a little valley,

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plant some corn, get a mill running, raise kettle, build a small house, and live your life.

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One can find traces of old settlements along almost any small creek, and the common pattern is that they have been abandoned at some point.

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One cause was the Great Depression that forced many people to move into the cities to find work. But whatever the cause, the fact that there are so many abandoned places paints a picture quite different from the often claimed steady progress, and thus of different times to come.

New inhabitants are ready to move in any day.

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A good example for all this is the Cave River Valley Natural Area, close to Spring Mills State Park, where today’s pictures are from. It’s story will continue next week.

The Desargues Configuration – A Quick Tour

Configurations are a wonderful way to confront the Euclideanly prejudiced with the abstraction of incidence geometries. Take for instance the Desargues configuration. Its ten points are given by the 10 different subsets with two elements of the set {1,2,3,4,5}. Here they are:


Its ten lines are given similarly by the 10 different subsets with three elements of the set {1,2,3,4,5}. Here they are, likewise:


Then we say that a point is incident with a line if is a subset of the line. For instance, the point {1,2} is incident with the three lines {1,2,3}, {1,2,4}, and {1,2,5}. Clearly, every point us incident with exactly three lines. Likewise, every line is incident with precisely 3 point: For instance, {1,2,3} is incident with {1,2},{1,3}, and {2,3}. This makes the Desargue configuration a configuration of type (10,3).


Above is a nice geometric interpretation of this. Take five planes in space so that no two of them are parallel and no three intersect in a single line. Then these planes will intersect in the 10 lines and 10 planes of the Desargues configuration. Unfortunately, there is no way I can think of to place five planes into space so that this will look pretty, the reason being that the “faces” of this configuration are nonconvex quadrilaterals. But one can do differently. Still using space, consider a tetrahedron. Take as points the four vertices of the tetrahedron as well as the 6 midpoints of the edges. Lines are the six edges of the tetrahedron and the incircles of the triangles. That requires bending our understanding of lines a little, but now this configuration becomes easy to visualize.

Desargue tetra

Where have the five planes gone? You could say that four of them have become the faces of the tetrahedron, and the fifth a wisely chosen sphere. The right image shows in a top view how to label the points and lines. If this is still too concrete, you can also use the following 2-dimensional drawing.


This is illustrating Desargue’s Theorem: Take two triangles p1, p2, p3 and q1, q2, q3 which are in perspective, i.e. have the three lines lines p(i)q(i) pass through a common point p. Now consider the intersection r12 of the two lines p1q1 and p2q2, and likewise the intersections r13 and r23. Desargue tells us that these three points lie on a common line. This sounds complicated, but given that we are only using the most elementary notion from geometry, namely incidence, it is surprising enough that there are non-trivial theorems at all. Why is this true? A miracle? Let’s move back into space and pretend for a second that the two triangles lie in different planes.

Desargues space

Then these planes meet in a line, which is the one through r12, r13, r23, because these points are intersections of lines that belong to one of the two planes each. So, in space Desargue was evidently right. Can things go wrong in the plane? Yes and no. No, because a Euclidean plane can always be though of as a plane in space, and we can understand any planar Desarguesian construction of the projection of a similar configuration from space. No, because there are in fact “planes” that do not lie in any space and where Desargues’ theorem is false.

The Grass is Still Growing

I have written about Columbus (Indiana) before. The little town cultivates a lot of modern architecture, given its size and location. This Fall it houses an exhibit of contemporary sculptures.

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Wiikiaami by studio:indigenous made me think of being caught in a gigantic fish trap (incidentally, the German word Reuse for it seems to have no english counterpart). At the moment, spiders have taken it as their new home.

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The Moore sculpture is now framed by Conversation Plinth from IKD.

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My favorite sculpture is Anything can happen in the woods by Plan B Architecture & Urbanism.

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They consist of metal columns seemingly growing next to grass covered mounds that were intended for sitting but are more used for climbing.

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Hyperbolic Inversion (Inversion II)

The first time I taught hyperbolic geometry, I thought I could be done in a week. I also thought that it would help to do spherical and hyperbolic geometry side by side. To save time, I did it not side by side, but simultaneously, using the quadratic forms
For negative ε, the spheres become hyperboloids, but the formula for inversion I(x) = x/ x ‧ x still works, with all its suitably formulated properties. Hence one can use the stereographic projection both for spheres and the hyperboloid in Lorentz space to get all the models with its simultaneously.

Needless to say, this was a complete disaster. Let’s just study the 2-dimensional hyperbolic inversion. Circles in this geometry are the level sets of the quadratic form above, which are hyperbolas. Below you see concentric hyperbolas with growing radius. For each radius, you get two branches. The squared radius can be 0, in which case you get the two diagonals, and even negative, so that the hyperbolas open upward and downward.

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These and their translates are the only hyperbolas we will consider, others belong to other quadratic forms. Let’s begin simple and invert a family of parallel horizontal lines at the blue hyperbola. Their images are hyperbolas with one branch intersecting the mirror hyperbola in the same points as the lines, and all touching at the origin.

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In fact, all lines are mapped to hyperbolas. Below are to families of line segments and their image hyperbolas.

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If you want to check this with a computation: A line through (a,b) making angle φ with the x-axis is mapped to a hyperbola with “center” p and squared radius


Then, certain hyperbolas (namely level sets of our quadratic form) are mapped to hyperbolas. Below, the purplish hyperbolas are concentric, and their greenish images pass through the same pair points. (Where do these intersections come from – – -?).

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In formulas: A hyperbola with center (a,b) and radius r is mapped to another hyperbola with


Are circles mapped to circles?

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Evidently not. Euclid isn’t here anymore, circles are not round, and we better don’t mention the rest.