Tanya Khovanova's Math Blog

I use Escher’s tessellations to teach wallpaper groups. Escher is the best painter among mathematicians and the best mathematician among painters. His fame helps energize my class. Plus, he has so many beautiful drawings to choose from.

However, there is another layer to his tessellations. Many paintings are not just a study in wallpaper groups but also in group-subgroup pairs. For example, consider these red/gray/black fishes on the left. There are three distinct points where three different reflection lines intersect. The first point is where three black fishes kiss each other. The other two points correspond to gray and red fish kisses. In orbifold notation, this symmetry group is *333.

But, the same drawing has another symmetry group. We just need to ignore color. That is, we consider all the fishes to be the same. In this case, our three distinct points where three reflection lines intersect become the same point: the point where fishes kiss, regardless of their color. The new symmetry group has an additional element: a 120-degree rotation where three fins of three different-colored fishes touch each other. Thus, the new symmetry group is 3*3.

Escher created a lot of examples of groups and their subgroups using color. But, sometimes, he was more subtle. In one of my previous posts, The Dark Secret of Escher’s Shells, I discussed my favorite Escher’s plane tessellation. In that drawing, the second group appears when we ignore the markings on one of the dark shells.

Here is another spectacular example of a group and subgroup, a tessellation of a hyperbolic plane with angels and devils. Do you see two different symmetry groups in the painting?

A Whitney umbrella is a cool surface I wanted to crochet. The umbrella continues to infinity, and there is no way I want to crochet the whole thing. I wanted to make a finite portion of a Whitney umbrella surrounding the most exciting point.

The result is seen in the picture. Technically, I crocheted not Whitney umbrellas but topologically equivalent surfaces. I am most proud of my secret — and painful — method of crocheting the self-intersecting segment. One day I will spill it.

As Wikipedia defines it: the Whitney umbrella is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line parallel to the parabola’s axis and lies on its perpendicular bisecting plane. If you look at the picture, the fixed straight line is the self-intersection line, which is a continuation of the line segment where the colors are woven through each other. You can find the parabola as the curve formed by the two-colored edge on either side of the umbrella. Oops, I forgot that only three of these umbrellas are made of two colors.

The Whitney umbrella is a ruled surface, meaning that for every point, there is a straight line on the surface that passes through the point. A ruled surface can be visually described as the set of points swept by a moving straight line.

Oh, look, the stitched rows can pretend to be these straight lines. Actually, if I fold these thingies, the stitched rows ARE straight lines. But, when I make the edges into parabolas, the rows stop being straight. In the real Whitney umbrella, if you look along the intersection line, the straight lines are closer to each other than they are along the parabola. But in crochets, the distances between rows have to be fixed. If my crochets are folded, they become rectangles and ruled surfaces. The real Whitney umbrella does not fold into a plane.

The Whitney umbrella is famous for being the only stable singularity of mappings from R² to R³. I am grateful to Paul Seidel for emailing me the proof. This singularity is so famous it even has two names: pinch point and cuspidal point. Though my crochets are not exactly Whitney umbrellas, they show this singularity. Hooray! I found a secret way to crochet the most famous stable singularity!

My favorite of the Escher plane tessellations is the one with shells. It is stunning, and the mathematics behind it is beautiful. I want to thank the late John Conway for teaching me the secret of this drawing.

Mathematicians are interested in tessellations because of the symmetries behind them. This tessellation has translational and rotational symmetries. Can you find them?

When I ask my students to find the rotational symmetries, they immediately tell me that they see two different 4-fold points, aka points where 90-degree rotations preserve the drawing. One point, I call G, is where four greenish shells meet, and one point, I call R, is where four reddish shells meet.

As you might have guessed, the students’ answer is not quite correct. There is more to the picture. Look at a dark-brown shell that looks like a curvy rectangle. This shape has markings. Now look at a specific point R and its four closest brown shells. You can see that going around this point R, the brown shells alternate their orientation: the darker side of these shells either faces towards point R or away.

The big secret of this artwork is that it contains TWO symmetry groups: a group and a subgroup. If we ignore the markings on the brown shells and consider them one solid color, then point R is indeed a 4-fold symmetry point. In addition, the center of the brown shape is a 2-fold symmetry point. Thus, the symmetry group of this simplified drawing is 442 in orbifold notation.

If we take the markings of the brown shells into account, then point R is not a 4-fold rotation, it is a 2-fold rotation. Point G keeps the property of being a 4-fold rotation. If you know your symmetry groups, you can conclude that there should be another 4-fold rotation. But where is it?

I will spill my answer. The symmetry point G is not ONE symmetry point anymore. There are two different points where greenish shells meet. The dark side of the brown shells faces one of them and looks away from the other.

The dark secret of this drawing is that it demonstrates two symmetry groups: group 442 and its subgroup 442, with different fundamental regions. To see the secret, you must look closely at a dark brown shell and find its darker side.

I do not like making objects with my hands. But I lived in Soviet Russia. So I knew how to crochet, knit, and sew — because in Russia at that time, we didn’t have a choice. I was always bad at it. The only thing I was good at was darning socks: I had to do it too often. By the way, I failed to find a video on how to darn socks the same way my mom taught me.

Then I came to the US. I suddenly found myself in a rich society, where it was cheaper to buy new stuff than to spend the time doing things with my hands. So I happily dropped my craftsmanship.

After not working with my hands for 28 years, one day I needed hyperbolic surfaces for my classes and I couldn’t find any to buy. Hyperbolic surfaces are famous for providing an example when the Euclid’s Fifth axiom doesn’t work. These hyperbolic surfaces look flat locally, so you can continue a line in any given direction. If you draw a line on such a surface and pick a point that is not on the line, then you can draw many lines through the point that are parallel to the given line.

My students are more important than my dislike of crochet, so I decided to just do it myself. I asked my friend Debbie, who knows how to crochet, for advice, and she gave me more than advice. She gave me a hook and a piece of yarn and reminded me how to work the hook. She started me with a small circle. After that all I had to do was add two stitches for each stitch on the perimeter of the circle. The finished product is this green ballish thing that looks like a brain, as in the photo.

Outside the starting circle, each small surface segment of this “brain” looks the same, making the “brain” a surface of constant curvature.

I chose a ratio of 2 to 1, adding two new stitches for each previous stitch. With this ratio, my flattish surface started looking like a ball very fast. The length of the perimeter doubled for every row. Thus each new row I crocheted took the same total amount of time that I had already spent on the whole thing. All the hours I worked on this “brain,” I kept thinking: darn, it is so unrewarding to do this. Annoying as it was, the thing that kept me going was my initial decision to continue to use up all the yarn Debbie had given me. In the end, with this ratio, half the time I worked was spent making the final row.Share:

The Fomenko drawing on the left is from the original Russian edition of Homotopic Topology by Fuks, Fomenko and Gutenmacher. Dmitry Fuchs signed this book for me after my success in the USSR Math Olympiad when I was in the 9th grade. For many years I didn’t know what the picture meant and was mystified by it. Now the book has been republished with explanations and is available in English at a non-affordable price. You can find this picture and many other Fomenko drawings in his book called Mathematical Impressions, which is affordable, although the comments accompanying the illustrations are confusing. So I have my own explanation for the meaning of this illustration.

The bracelet is made out of shells. Each shell is a hollow cone whose vertex is glued to a point on the rim of the cone’s opening, thus giving each hollow cone its own handle. In a part of another drawing (at left), Fomenko shows how the bracelet is built by an army of tiny slaves. First they build the shells and then they connect them together.

How do they connect the shells to each other? The rim of the next shell is glued to the handle of the previous shell. Let me remind you that a straight line connecting a point on the rim to the vertex of a cone is called a generatrix. Imagine a generatrix that connects a vertex of a cone to the point on the rim to which this vertex is glued. This generatrix becomes a circle in a shell, which I call the handle circle. So the rim of the next shell is glued to the handle circle of the previous shell.

Now consider the fundamental group of a shell. The rim can be contracted to the handle circle. Moreover, the cone itself can be contracted to the handle circle. If we glue several shells together, the result is contractible to the handle circle of the last shell.

Now let’s go back to the bracelet. The shells become smaller in both directions and end in two points. The front end point is more interesting topologically than the one in back. Every point other than the front end has a contractible neighborhood, while the front end point does not. Or in scientific terms: The bracelet gives an example of a space with a point at which the space is “1-lc” but with no open neighborhoods on which every (Cech) 1-cycle bounds.Share:

This fractal was designed by Ross Hilbert and is named “Weathered Steel Weave.” You can find many other beautiful pictures in his fractal gallery.

The fractal is based on iterations of the following fractal formula z_new = cos(c z_old), where the Julia Constant c is equal to −0.364444444444444+0.995555555555556i. To produce the image, you need to start with a complex value of z and iterate it many times using the formula above. The color is chosen based on how close the iteration results are to the border of the unit circle.Share:

Is there a way to put a sequence of numbers to music? The system that comes immediately to mind is to match a number to a particular pitch. The difference between any two neighboring integers is the same, so it is logical to assume that the same tone interval should correspond to the same difference in integers. After we decide which tone interval corresponds to the difference of 1, we need to find our starting point. That is, we need to choose the pitch that corresponds to the number 1. After that, all numbers can be automatically matched to pitches.

After we know the pitches for our numbers, to make it into music we need to decide on the time interval between the notes. The music should be uniquely defined by the sequence, hence the only logical way would be to have a fixed time interval between two consecutive notes.

We see that there are several parameters here: the starting point, the pitch difference corresponding to 1, and the time interval between notes. The Online Encyclopedia of Integer Sequences offers the conversion to music for any sequence. It gives you freedom to set the parameters yourself. The sequences do not sound melodic because mathematical sequences will not necessarily follow rules that comply with a nice melody. Moreover, there are no interesting rhythms because the time interval between the notes is always the same.

One day I received an email from a stranger named Michael Blake. He sent me a link to his video on YouTube called “What Pi Sounds Like.” He converted the digits of Pi to music. My stomach hurt while I was listening to his music. My stomach hurts now while I am writing this. He just numbered white keys on the piano from 1 to 9 starting from C. Then he played the digits of Pi. Clearly, Michael is not a mathematician, as he does not seem to know what to do with 0. Luckily for him the first 32 digits of Pi do not contain zero, so Michael played the first several digits over and over. My stomach hurts because he lost the basic math property of digits: the difference between the neighboring digits is the same. In his interpretation the digits that differ by one can have a tone interval of minor or major second in a random order corresponding to his random starting point.

I am not writing this to trash Michael. He is a free man in a free country and can do whatever he wants with the digits of Pi. Oops, I am sorry, he can’t do whatever he wants. Michael’s video was removed from YouTube due to an odd copyright infringement claim by Lars Erickson, who wrote a symphony using the digits of Pi.

Luckily for my readers Michael’s video appears in some other places, for example at the New Scientist channel. As Michael didn’t follow the symmetry of numbers and instead replaced the math rules with some music rules, his interpretation of Pi is one of the most melodic I’ve heard. The more randomly and non-mathematically you interpret digits, the more freedom you have to make a nice piece of music. I will say, however, that Michael’s video is nicely done, and I am glad that musicians are promoting Pi.

Other musicians do other strange things. For example, Steven Rochen composed a violin solo based on the digits of Pi. Unlike Michael, he used the same tone interval for progressing from one number to the next, like a mathematician would do. He started with A representing 1 and each subsequent number corresponded to an increase of half a tone. That is, A# is 2 and so on. Like Michael Blake he didn’t know what to do with 0 and used it for rest. In addition, when he encountered 10, 11, and 12 as part of the decimal expansion he didn’t use them as two digits, but combined them, and used them for F#, G, G# respectively. To him this was the way to cover all possible notes within one octave, but for me, it unfortunately caused another twinge in my stomach.Share:

This is a version of the standard charades game that my son, Sergei Bernstein, invented.

Unlike in regular charades, the person who acts out the phrase doesn’t know what the phrase is and has to guess it. The viewers on the other hand, know the phrase but they are not allowed to talk.

So the actor is blindfolded and the viewers are not just watching; they are actively moving the actor and his/her body parts around to communicate the phrase. For example, if the actor is on the right track, since the viewers can’t say, “Yes, good!”, they might communicate it by nodding the actor’s head.

Sounds like fun, especially for people who enjoy touching and being touched.Share:

You might ask why this piece is titled George Hart, when the only man in the photo on the left is John Conway. George Hart is related to this picture in three different ways.

First, this picture is of the math department common room at Princeton University. It was taken during a joint event of the WaM and SWIM programs in June, 2009. It shows the Zometool workshop conducted by George Hart that resulted in the construction of the expanded 120-cell, which appears in the photo’s foreground.

The second connection to George Hart is that beautiful shiny object under the lights on the far left. The object is the propello-octahedron sculpture that George Hart created out of 150 CDs. The sculpture has been in the common room for many years, and I have always loved it.

Unfortunately, the sculpture was slowly degrading, even losing some of its parts. I visited Princeton in August 2008 and realized that the sculpture was facing a short life expectancy, so I took the picture of it that is below. I couldn’t find any angle to shoot the photo that hid the lost pieces. The sculpture survived until my visit in June 2009, as evidenced by the first picture. But unfortunately it wasn’t there any more during my last visit in May 2010.

Oops, I almost forgot. I promised you a third way in which George Hart relates to the first picture. He is the one who took it.Share:

I am used to thinking that a “woman in numbers” means a female number theorist. But not anymore. I just discovered drawings by Svetlana Bogatyr. From now on the expression a “woman in numbers” will convey an additional meaning to me.

I am grateful to Svetlana for permitting me to post several of her drawings. The “Mature Woman” is on the left. “Eurydice”, “Girl in Scarf” and “Holland Woman ” are below.

Enjoy.

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