Inversion at the Circle

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My first encounter with the inversion at a circle happened through Stan Ogilvy’s wonderful little book Excursions in Geometry. It is like a magic key: When you know how to use it, it opens many doors. The definition is simple: A point at distance r from the center of the uniut circle is inverted to a point on the ray through that point at distance 1/r.

One of the magic facts one has to learn is that this inversion maps lines or circles to lines or circles. More precisely:

  1. Lines through the origin or mapped to lines through the origin.
  2. Circles through the origin or mapped to lines not through the origin (and vice versa).
  3. Circles not through the origin or mapped to circles not through the origin.

Why is this true? One can of course prove this by computation. The formulas are not pretty and provide little insight. One can argue geometrically, but the diagrams get very messy. I suspect that a decent way to to this is to define a circles/line through three points as the set of points whose cross ratio with the three points is real, and then show that Möbius transformations preserve cross ratios. But arguments quickly become circular this way…

Here I am outlining a hybrid approach. We will only use two simple facts about triangle. Thales theorem

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characterizes circles as those points that make a right angle over a given hypothenuse. Secondly a triangle ABC has a right angle at C if the pre-Pythagorean theorem AD AB = AC^2 holds. This follows from the similarity of triangles ABC and ACD.

We will now begin with the simplest non-trivial case, where the vertical straight line tangent to the unit circle at 1, is inverted to a circle with center at 1/2 and radius 1/2.

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We use our geometric preliminaries: By Thales, the point q’ lies on the orange circle iff the angle at q’ is a right angle. This is the case iff the two triangles 01q’ and 0q1 are similar, and that is true iff |q||q’|=1.

That case was easy. All other non-trivial cases have significantly more complicated proofs, as far as I know. Therefore we will reduce all other cases to the one we just saw.

We will look at an arbitrary vertical line L not through the origin.
Let R(z) = r z be a homothety with r being fixed positve number chosen so that R maps our line to the vertical line through 1. We denote the inversion by I(z) = z/|z|^2. Please check in your head that R I R = I. This means that we can determine I(L) by computing R(I(R(L))). This is easy, because R(L) is the vertical line through 1, I(R(L)) the circle centered at 1/2 with radius 1/2, and R(I(R(L))) a circle through the origin. Because inversion is compatible with rotations about the origin, we have no proven the second statement above. The first was trivial.


Next we look at a circle C that touches the unit circle at the point 1.
For notational simplicity, we will argue with the reciprocal map 1/z instead of I(z). They only differ by a complex conjugation. Then

  • z-1 maps C to a circle through 0.
  • 1/(z-1) maps C to a line not through 0.
  • 1+1/(z-1) maps C to another vertical line.
  • 1/(1+1/(z-1)) maps C to another circle.
  • 1-1/(1-1/(1-z)) maps C to another circle.

So far so good. That last expression turns out to be equal to 1/z, so that in fact 1/z maps C indeed to a circle. I promised no serious computations.
Why then is this strange identity true? By taking reciprocals, it can be written even more dramatically as


where φ(z) = 1/(1-z). No φ is a Möbius transformation that maps 0 to 1 to ∞ to 0. This means that its third power fixed these three points and therefore must be the identity as claimed.


Lastly, we need to consider a general circle not through 0. Using a homothety R as before, we can map it to a circle touching the unit circle, invert, and scale again, using RIR=R.

It is a pity that this material is not taught in high schools, and even considered “obscure”. Its close cousin, the stereographic projection, and its corresponding properties were at the foundation of Astronomy and Navigation all the way through the Middle Ages. Only Mercator’s projection improved on the Astrolabe. Moreover, the inversion opens the door to hyperbolic geometry.


In 1999, I had my first chance to witness a solar eclipse. That was in Bonn, Germany, and only a partial eclipse. It was very partial, because the sky was cloudy.

Now, in 2017, I didn’t feel like driving for three hours to get stuck in a traffic jam. So instead I contented myself with another partial eclipse and went to Brown County State Park.

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The view from the fire tower was a little eery because the sky was significantly darker. Capturing the eclipse with a wide angle lens is a little silly, but safe for eye and camera.

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Using a tele lens is danegerous, never look through the camera at the sun even with a strong neutral density filter. I used a 10 stop filter on a 300mm lens. This turned out not be quite enough to darken the sun, but one can now at least see the eclipse (and it didn’t fry my camera).

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My main interest, however, was how my favorite lakefront at Strahl Lake that I have photographed (too?) many times would look like during the eclipse.

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Parking Garages

The first examples of periodic minimal surfaces with helicoidal ends (besides the helicoid itself) are Hermann Karcher’s twisted Scherk surfaces from 1988.
Here are a few of them, rendered with Bryce back in 1999.


As you can see, these can be twisted more and more so that they appear to become two helicoids glued together. In this case, the two helicoids turn the same way. A few years later, Martin and I were looking at more general ways of gluing helicoids together to obtain new minimal surfaces. The model case is what we called a parking garage structure: You can describe them mathematically as superpositions of complex argument functions, like so:


Here the numbers z(k) designate the location of the axis viewed from above, and the ε(k) can be +1 or -1, depending on the spin of the helicoid.
Note that the graph of the multivalued function arg(z) is half of a helicoid (that stays on one side of the vertical axis).

An example with three helicoidal columns of the same spin, placed at -1, 0, and 1, looks like this:


If you alternate the spin, you get surfaces that untwist to higher genus helicoids, we believe.


It is also possible to place the columns off a common line, like so:


Nobody knows what minimal surfaces these untwist to.

The images above were made with Mathematics in 2001. Later I found a simple way to do this in PoVRay, which I might explain next time. Here an image from 2002:

1 3 a slice 025 m

Most people get easily lost in parking garages that have only two columns. It would be cool to have a computer game where one can walk around these more complicated structures, with the location of the columns moving in time …

Old And New (Summer in Berlin III)

There are many good places to contemplate the clashes between old and new in Berlin, and one of them is the area along the Spree near the U-Bahn station Schlesisches Tor. This is where the world ended for people living in West-Berlin while the city was divided. Now one can walk across the bridges and admire the construction circus on both sides.

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Herbert George Wells might have thought that his phantasies have come true. When they are done with all this, will it looks like this?

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And will we get more playful little sculptures like the Molecule Man by Jonathan Borofsky?

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There is some obvious resistance. It feels like the perfection of a finished building is stifling the creativity.
Who wouldn’t want to defend the octagonal brick building below?

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Do we really want to lose all this?

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My taste is more for blending old and new and let them coexist.

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Magnetism is played by two players on a strip of squares, who take turns placing + and – tokens onto the strip. The only rule is that no two tokens with the same parity can be placed next to each other. For instance, there are three legal moves in the following position:


The player who moves last, wins. This makes Magnetism an impartial game, so that each position is equivalent to a Nim-pile. It turns out that Magnetism is very simple.
First we notice that any position is the sum of simpler positions that have tokens just at the end of a strip. (A sum of games is played by first choosing a game summand, and then making a move in that summand).

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Therefore we will know everything about Magentism if we can determine the size of the Nim-piles (the “nimbers”) of the 9 elementary positions:

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Things get even simpler. Because of the symmetry of things, there are only four truly different boards to consider.

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Let’s denote the nimbers of a board of n empty squares (thus not counting the tokens at the end when present) by G(n), G+(n), G++(n), and G+-(n).

n 0 1 2 3 4 5 6
G(n) 0 1 0 1 0 1 0
G+(n) 0 1 2 3 4 5 6
G++(n) 1 1 1 1 1 1
G+-(n) 0 0 0 0 0 0 0

Now you can win in a position with a positive nimber by moving to a position with zero nimber. For instance, on a board with a single + at one end, one possible winning move is to put a – at the other end.

The Door Was Open (Summer in Berlin II)

I like architecture, or, to be precise, certain states of buildings. Ruins are fascinating, but even more so construction sites. Both are usually off limits (as are the corresponding states of human affairs, death and conception, unless you are involved one way or the other). So I am often forced to trespass a little.

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In this case, as you can see, the door was open, and I just couldn’t resist.

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Views like the one above make it instantly clear that we are not on a generic construction site. Somebody with taste has been designing this, and whoever is doing the construction work, is doing an excellent job by creating crystal clear previews of what’s to come.

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Wondrous tools are on display too, just for me. I can only guess their purpose by looking at the ornamented concrete slabs. Everything is purposeful, even the occasional leftover tile.

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What fascinates be most at places like these is the tension between the clarity of the present and the vagueness of an undefined future.

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Curved is also Beautiful

Among the many helicoids with handles, the translation invariant genus one helicoid is by far the simplest. It was first constructed by David Hoffman, Hermann Karcher, and Fusheng Wei. You can learn almost everything about it from a single image.

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The right hand side is a portion of the actual minimal surface, which extends by rotations about its horizontal and vertical lines to the complete surface.


The quotient of this surface by its vertical translation is a torus, and the presence of the two straight symmetry lines hint that this is a rhombic torus, which you see outlined black in the left top left image. Its two diagonals become the two straight lines of the surface. The trick is to see the surface patch to the top right as the image of the colorful rectangle on the top left. The top left and bottom right corners of that rectangle are bent together so that they touch, the horizontal edges align as the horizontal line, and the vertical edges align as the vertical line of the surface.

The two semicircular arcs become the half turns of the two helicoidal arcs, this allows to truncate the surface image nicely. The mesh lines of the colored rectangle are, incidentally, obtained by conformal mapping a rectangle to itself in a quirky way:

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Here, the vertical edges of the left rectangle are mapped to the two semicircles, and the horizontal edges to everything else. The “extra” vertical lines are included so that we hit all vertices of the right rectangle by a parameter line.

So that is all very easy. The tricky part is make the right choices in order that the the two opposite corners of our parameter rectangle really meet. The horizontal alignment is achieved by using as a rhombic torus the funny 70.7083 degree rhombus we discussed last time. If you choose another rhombus, the two verical line segments will not match up.

To guarantee also a vertical alignment of the two corners, one needs to choose the location of the two points E1 and E2. To do this, one constructs a meromorphic 1-form on the torus which has simple poles at E1 and E2 and two zeroes at V1 and V2 (whose location depend on E1 and E2 by Abel’s theorem). The integral will map our colored rectangle to a slit domain consisting of two merged half strips. The ends of the half strip correspond to the two helicoidal ends of the surface.

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That the two slits line up in this picture is no coincidence. E1 and E2 have been chosen so that this happens (thanks to the intermediate value theorem). Tt is exactly what is needed to achieve the vertical alignment of the two corners.

The Jewish Museum (Summer in Berlin I)

Berlin has changed a lot since the wall came down in 1989. Most notably the constricted architecture from before finds its counterpoint in buildings that show a liberated sense of what can be done with space.

One of my favorites is the Libeskind addition to the Jewish Museum from 2001.

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You can only enter it underground and are confronted immediately with long and slanted corridors.

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I felt the natural way to photograph this is by slanting the camera as well. There is a lot of narrow vertical space,

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admitting just enough light so that we don’t feel claustrophobic.

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Then there are the Voids, most of them inaccessible, but present through views and gaps in our perception.

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We lose the distinction of being inside or outside, but we learn that is us who create the space around us.