“Unusual Geometries” at Oklahoma State.

A posting at OKState News on October 17, 2023 announced the opening of “Unusual Geometries,” an exhibition at the Oklahoma State University Museum of Art (runs until December 16). The show promises its guests “the hidden beauty and playfulness found within the world of geometry.” Among the art on display is an image with the title “Cohomology fractal for the SnapPy manifold s227,” by David Bachman (Pitzer College, Claremont), Saul Schleimer (Warwick) and Henry Segerman (OKState).

A blue and white fractal pattern.
David Bachman, Saul Schleimer and Henry Segerman, Cohomology fractal for the SnapPy manifold s227. Image courtesy of Henry Segerman.

What does this picture show? “SnapPy manifold s227” is the name of a certain mathematical object: it’s a finite-volume three-dimensional space with a hyperbolic metric. The authors have put together a YouTube video explaining in elementary terms what these words mean, and how the image corresponds to a cohomology class. They work their way up from two dimensions, where things are much easier to visualize. There we talk about area instead of volume. An example of a 2-dimensional space with finite area is the sphere, or more interestingly, the torus (a donut shape).

a. A flat plane, b. A donut shape.
a. The plane is the standard 2-dimensional space; it has infinite area. b. The torus is an example of a 2-dimensional space (near any point you could think you were in the plane) with finite area.

One way to get a (1-dimensional) cohomology class on the torus is to start with a closed curve on the surface, like the black one in the figure below. Give it an orientation: decide which direction along the curve you will count as “positive.” The arrow on the black circle shows which orientation is chosen in this example. Then pick any other closed, oriented curve on the torus. You can assign to this second curve a whole number by summing the intersection numbers of the two curves. Each intersection contributes either $+1$ or $-1$ depending on what direction the curves are headed when they cross each other, as illustrated in the figure. The intersection sum associates a whole number to each closed oriented curve on the torus. This association is the cohomology class.

A donut shape. A black circle threads through the donut hole and around the outside. A red loop sits on the outside of the donut, intersecting the black circle at two points. A green curve winds twice around the entire shape, threading through the hole once. To the right, a diagram shows that when a black arrow pointing upwards intersects with a blue arrow, it adds +1 if the blue arrow points right, and –1 if the blue arrow points left.
The black curve, like any closed, oriented curve on the torus, defines a 1-dimensional cohomology class as described above. Its value on the green curve is $-2$, and on the red curve, $0$.

Bachman, Schleimer, and Segerman’s image is of the inside of a three-dimensional space. There the analogue of a closed curve is a closed, oriented surface, and like our black curve this surface defines a 1-dimensional cohomology class. Their space is hyperbolic, which means that it has constant negative curvature. In two dimensions, such a space would be saddle-shaped everywhere. The area of a circle around such a point increases more rapidly with the radius $r$ than the $\pi r^2$ we’re used to in the plane. If every point is a saddle point, this area blow-up makes a finite-area surface of constant negative curvature impossible to realistically portray on the page, even though there are many of them.

A saddle shape.
In two dimensions, a point of negative curvature is a saddle point.

In their video, Bachman, Schleimer, and Segerman ask you to imagine that you are living in this space, and that when you look through that surface in the positive direction, everything behind it looks darker; whereas looking in the negative direction makes everything appear lighter. The picture on exhibit is what you would see.

Additionally, the authors have an expository paper about the construction, written for the participants in the interdisciplinary Bridges Conference on math and art, 2020.

Numbers in the human brain.

A news item in Nature titled “Your brain finds it easy to size up four objects but not five — here’s why” highlights an October 2 paper from Nature Human Behaviour. Nature staff writer Mariana Lenharo reaches back in the Nature archives for early research on the topic. In Volume 3 (1871) W. Stanley Jevons published “The Power of Numerical Discrimination”, including the table replicated here.

A chart showing Jevons' estimated numbers vs. actual numbers of black beans thrown. When the number is 3 or 4, his estimates are all correct. As the numbers get bigger, the spread of estimated numbers get wider, and the proportion of mistakes generally increases.
Jevons ran 1027 trials on himself, casting a randomly grasped handful of black beans into a white paper box, and recording both his instant estimation of the number thrown and what it actually was. For example, in the 156 trials involving seven beans, he “saw” six 18 times, seven 113 times, and eight 25 times. Reproduced from Jevons, W. “The Power of Numerical Discrimination.” Nature 3, 281–282 (1871).

As Jevons’s experiment shows, our ability to grasp immediately how many objects we are faced with (called subitizing) works up to four items and becomes more and more unreliable afterwards. Lenharo tells us how the authors of the Nature Neuroscience article, a group of German epileptologists and physiologists led by Florian Mormann (Bonn) and Andreas Nieder (Tübingen), were able to improve on Jevons by using a larger number of subjects (17) and technology that he likely could never have imagined. For instance, Mormann and Nieder’s group had access to patients with microelectrodes implanted in their brains. This allowed the team to monitor the activity of individual neurons. It has been known for some time that single neurons in the primate brain can encode numbers of items (a review article published in 2009 by Nieder and Stanislas Dehaene cites work dating back to 2002). In the new experiment, subjects were asked to guess whether the number of dots shown on a screen was odd or even while researchers tracked neural activity.

The new study shows that the neurons associated with the numbers 1-4 are different from the others. As Lenharo explains it, “neurons specializing in numbers of four or less responded very specifically and selectively to their preferred number. Neurons that specialize in five to nine, however, responded strongly to their preferred number but also to numbers immediately adjacent to theirs.” She quotes Nieder: “The higher the preferred number, the less selective these neurons were,” giving a nice cellular-level perspective on Jevons’s experimental results. The article’s abstract ends with the suggestion that this difference may be connected to “capacity limitations” in our attention and working memory. These limitations are probably familiar to everyone: trying to drive and eat an ice-cream cone while discussing relativity is not a good idea.

Polymer helicity and knot chirality.

First the definitions.

A spiral curve in 3-space has right or left helicity. Right helicity is the same kind as a standard corkscrew: when you turn it to the right (clockwise) it goes forward. For a point on the curve, moving forward (in either direction) means rotating clockwise. The mirror image of a curve with right helicity is a curve with left helicity. In plumbing, screws with this structure are called left-threaded or reverse-threaded.

Left: A corkscrew that twists counterclockwise as it moves forward in space. Right: Corkscrew that twists clockwise as it moves forward in space.
Left (red) and right (green) helicity.

When you take the mirror image of certain knots, their over- and under-crossings get interchanged: Two threads that cross one another will switch positions. The thread that was on top will now be on the bottom, and vice versa. This is a topologically different object; these knots can’t be deformed into their mirror images. These knots are said to have chirality (literally, handedness): they exist in a left-handed and a right-handed form.

Three pairs of knots. The knots of each pair are identical, except that over-and under-crossings are switched.
Left-handed (red) and right-handed (green) forms of a. the trefoil knot ($3_1$ in the knot tables) and the two five-crossing knots b. $5_1$ and c. $5_2$. These knots are drawn in their open form: the two free ends of the string are so far away that they don’t enter into knotting or unknotting. Knots in molecules are usually of this type.

How is handedness calculated? The string bearing the knot is first given an orientation (it doesn’t matter which). Using the orientation, an index $\pm 1$ is assigned to each crossing in a 2D drawing of the knot: $+1$ if the rotation from the positive direction in the “over” strand to that in the “under” is counter-clockwise, and $-1$ if it is clockwise. If the sum of these indices is non-zero, the knot is chiral (has a handedness), left if it is negative, right if it is positive.

Three knots, with arrows of showing whether each crossing is under- or over.
The handedness calculation gives $-5$ for our left-handed $5_2$ and $+5$ for our right-handed $5_1$. In contrast, the figure-eight knot ($4_1$) gives $0$. It is in fact the same knot as its mirror-image: a $180^{\circ}$ planar rotation makes the projections coincide.

These related concepts appear in “Can Polymer Helicity Affect Topological Chirality of Polymer Knots?”, published in ACS Macro Letters (American Chemical Society) on January 27, 2023 and picked up in Nanowerk News on October 4. The authors, a group of physicists and chemists in Mainz led by Peter Virnau and Kostas Daoulas, used computer simulations to investigate the interplay when long polymer chains of one helicity or another (or none) spontaneously form knots. The results are summarized in this graph printed in their abstract. The strength of the helicity strongly predicts the handedness should a particular knot occur, except that there is a striking difference of behavior between the two five-crossing knots $5_1$ and $5_2$.

Plot showing probability of forming a right-handed knot as a function of strength of polymer helicity. Curves are plotted for knots $3_1$, $5_1$, and $5_2$. $3_1$ and $5_1$ get more likely to be right-handed as the polymer gets more right-handed; $5_2$ is the opposite.
For three kinds of knots ($3_1, 5_1$ and $5_2$), the probability of occurrence of the right-hand form is plotted against the helicity of the polymer chain. For $3_1$ and $5_1$, the probable handedness of the knot correlates with the helicity of the chain. Remarkably, the pattern for $5_2$ is counter-intuitive: the handedness of the knot varies oppositely to the helicity of the polymer. This image and the next from Zhao, Y. et al. “Can Polymer Helicity Affect Topological Chirality of Polymer Knots?” ACS Macro Lett. 2023, 12, 2, 234-240, licensed under CC-BY 4.0.

The authors explain the difference in behavior between $5_2$ and the torus knots $3_1$ and $5_1$ in terms of “braids” naturally occurring when the polymer crosses itself.

Coiled helices crossing one another in different configurations.
Using wire models of $5_1$ (a.) and $5_2$ (b.) knots produced in their simulations the authors illustrate how the same polymer helicity (right in these examples) can lead to opposite crossing numbers. The arrows show the orientation of the chain. Two strands going in the same direction, as in $5_1$, will lead to positive crossing numbers; whereas opposite directions, as in $5_2$, will lead to negative crossings.

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