May & June topics:

Non-commutative impossible objects.

In May, Scientific American ran “Impossible Objects: Mathematicians have designed a new kind of paradoxical shape.” The “impossible objects” in question exist only on paper. They look like perspective drawings of three-dimensional objects, but the various parts of the object cannot rationally be perceived as a whole. This new one is more than a curiosity: It has a mathematical life of its own.

The British physicist and geometer Roger Penrose designed the first impossible objects in 1954 after encountering similar visual paradoxes in M. C. Escher’s work. Penrose went on to give a cohomological formulation of the phenomenon (treated in a feature column on this site). Others took up that approach, including Robert Ghrist of the University of Pennsylvania. The most recent progress was achieved in this July 2025 preprint, co-authored with Zoe Cooperband. The impossible object highlighted in Scientific American, which makes novel use of non-orientability, derives from this work. Here we will explore a simplified version of the object, pared down to its mathematical core.

An image showing three platforms. The lowest is in the bottom left corner; a ladder reaches up to a second platform that stretches from the top left to the bottom right corner; another ladder reaches from the second platform to a third platform in the upper right corner. Arrows point clockwise along the rectangle's left, top, and right edges; an arrow points counter-clockwise along the bottom edge.
The new object is drawn on a Klein Bottle, the surface obtained by identifying the edges of this rectangle according to the arrows. Top is joined to bottom to form a cylinder, and right is joined to left after an up-down flip, as in the construction of a Möbius strip. Credit for this image and its variants below: Tony Phillips. They are loosely adapted from images in Scientific American.

In this simplified version, the new object has three flat levels joined by ladders. One ladder joins the first level to the second level, another joins the second level to the third. There are no paradoxes yet. The impossibility starts when we curl and twist the rectangle into a Klein bottle. To do this, we roll the drawing from top to bottom to get a cylinder. Then we give the right circular edge an up-down flip and glue it to the left one, so that the arrows along these edges end up pointing in the same direction.

Now, we have a paradoxical object. Following the top-down identification, one can walk directly from the first level to the second and the third. But following the twisted left-right identification, one can also walk from the first level directly onto the underside of the third level, and vice versa. These two paths cannot be reconciled into a 3D object.

As Moskowitz explains, this object has an unusual property, related to paths on the object and the order in which they are traversed.

Two paths in the object leading from a point back to itself are shown in color. The first one (a, blue traveler) goes from the first level, up a ladder to the second level and then directly back to the first, following the identification. The second (b, red traveler) goes from the first level, up two ladders to the third one and then directly back to the underside of the first.

What happens when the two paths are combined? The two combined paths a-then-b (a followed by b) and b-then-a  both end up in the same place—the underside of the left level—but the travelers’ perceptions of their final destination will be very different.

Path a-then-b. The traveler first follows path a in steps 1 to 4 and then (change color) path b in steps 4 to 9. During this route, the traveler climbs up three ladders, and feels that they are three levels higher than when they started.
Path b-then-a. In steps 1-6 the traveler follows path b, and climbs two ladders. Now on the underside of the first level, the traveler changes color and follows path a down the ladder to the underside of the second level, and then the rest of path a back to the starting point on the underside of the left level. The traveler has now gone up two ladders and down one, and so perceives that they have risen one level.

This is an example of non-commutativity: the path sequence a-then-b gives a different result from the sequence b-then-a. According to the authors, this is the first example of a non-commutative impossible object.

Non-commutativity is an algebraic concept. For a mathematical explanation of the difference between the effects of b-then-a and a-then-b, we have to introduce an algebraic structure. This is the fundamental group of a surface.

  • Fundamental group. Consider the set of all loops on the surface. Sort those loops into buckets consisting of those loops that can be continuously deformed one to the other. In topology, these buckets are called homotopy classes. The homotopy classes are the elements of the fundamental group. To make them into a group we need to define an operation, like the addition of numbers, that associates to any ordered pair $\alpha, \beta$ of elements a product $\alpha\beta$. We do this by choosing a loop a in the class $\alpha$ and a loop b in $\beta$, and defining $\alpha\beta$ to be the homotopy class of the loop a-then-b. It may be the case that $\alpha\beta = \beta\alpha$ for any two elements $\alpha$ and $\beta$; then the group is called commutative.

  • Equivalent curves. i. On the sphere, any curve can be shrunk down to a point. ii. On the torus, a-then-b can be deformed to b-then-a. iii. On the Klein bottle, a-then-b can be deformed to b-then-a$^{-1}$. Image credit: Tony Phillips.
  • Examples of fundamental groups. As illustrated in the above picture, any curve on the sphere can be shrunk to a point, so the fundamental group of the sphere is the “trivial” group with only one element $\varepsilon$. The fundamental group of the torus is generated by the classes $\alpha$ and $\beta$ of the curves a and b. Since as shown a-then-b can be deformed to b-then-a, it follows that $\alpha\beta = \beta\alpha$, so this group is commutative. The fundamental group of the Klein bottle is like that of the torus, except that now a-then-b is equivalent to b-then-a$^{-1}$, so $\alpha\beta = \beta\alpha^{-1}$. This group is not commutative.

The non-commutativity of the Ghrist-Cooperband impossible object is made possible by the non-commutativity of the fundamental group of the Klein bottle. On the other hand, the relation $\alpha\beta=\beta\alpha^{-1}$ suggests that the curves a-then-b and b-then-a$^{-1}$ should lead to the same experience for our travelers, and they do.

The traveler on route b-then-a$^{-1}$ first traverses path b (steps 1-6; at step 6, color change and direction reversal), then follows path a backwards (steps 6 to 9) on the underside; this will involve going up the ladder from the second level to the first. Just like the traveler on route a-then-b, this traveler experiences a three level increase in altitude.

What is happening here? To get a more accurate picture we need to understand a structure related to the fundamental group of the Klein bottle. This is the infinite dihedral group $D_{\infty}$. $D_{\infty}$ is the limit of the family of symmetry groups of regular $n$-gons. Like them, it has two generators: $r$ (rotate one step to the right) and $f$ (flip with respect to the basepoint).

Top: A triangle, square, and pentagon with their reflective and rotational symmetries. Bottom: An arc of a circle, with a line of reflection along the vertical and integers marking the distance from the line of reflection. The distance between 0 and 1 corresponds to a rotation r.
The infinite dihedral group $D_{\infty}$ has generators $r$ and $f$ just like the symmetry groups of regular $n$-gons. Image credit: Tony Phillips.

In all these groups the generators $r,f$ satisfy two relations. First, $f^2 = e$ (the identity), since flipping twice gets you back to where you started. Second, $rf$ ($r$ followed by $f$) is equal to $fr^{-1}$, i.e. rotate right and then flip has the same effect as flipping and then rotating left. For example $1\stackrel {r}{\rightarrow} 2 \stackrel {f}{\rightarrow} -2$ and $1\stackrel {f}{\rightarrow} -1 \stackrel {r^{-1}}{\rightarrow} -2$.

As we just discussed, the relation $rf=fr^{-1}$ also appears in the fundamental group of the Klein bottle. This resemblance can be made more explicit: By taking $\alpha$ to $r$ and $\beta$ to $f$, we get a map from the fundamental group of the Klein bottle to $D_{\infty}$ that respects the group product. Such a map is called a homomorphism. The mathematical meaning of the Ghrist-Cooperband impossible object is that it defines another, different homomorphism $h$ between these two groups.

Here’s how $h$ works. If we imagine our traveler starting at level $0$, then we can identify the perceived levels the traveler experiences with the vertices $\dots, -2, -1, 0, 1, 2,\dots$ of the infinite polygon. Going up one level corresponds to rotating one step to the right, and moving to the underside corresponds to flipping about $0$ (since three levels up is now perceived as three levels down). Now suppose we’re given a homotopy class $\gamma$ of loops on the Klein bottle. We pick a representative loop g, we send a traveler around g and record the experienced level change as an element of $D_{\infty}$. This will be $h(\gamma)$. So $h(\alpha)=r$ (up one level) and $h(\beta)=r^2f$ (up two levels and onto the underside). Because $h$ is a homomorphism, $h(\alpha\beta) = h(\alpha)h(\beta) = r^3f$ and $h(\beta\alpha) = h(\beta)h(\alpha)$ $= r^2fr = rf$ are different, so the non-commutativity of the fundamental group of the Klein bottle translates to the non-commutativity of the traveler’s experiences.

Note that for any integer $n$ the equations $h_n(\alpha) = r$, $h_n(\beta)= r^nf$ also define a homomorphism from the fundamental group of the Klein bottle to $D_{\infty}$. Exercise: Construct, for each $n$, an impossible object on the Klein bottle that defines $h_n$ just as the Ghrist-Cooperband object defines $h=h_2$.

Voronoi diagrams in Chinese money-plant leaves.

The veins in plant leaves have a variety of configurations, but the dominant mathematical models only address open, branching vein systems like the first three in the image below. “Your Houseplant Has Been Solving An Ancient Geometry Problem All Along,” on ScienceBlog.com, covers new work which changes that.

Four vein patterns. Three are branching, and the last is reticulate: It partitions the leaf into small disjoint regions.
Some of the configuration types that can occur in the vein system of a leaf. Picture credit: detail of Leaf morphology in Wikipedia, used under CC BY-SA 3.0 license.

The work appeared in Nature Communications on May 12. It focuses on the Chinese money plant Pilea peperomioides. The authors find that the veins in this plant are closely approximated by a purely geometrical structure called a Voronoi diagram, and they suggest a biological explanation for this.

Voronoi diagrams. A set $P$ of points $p_1, \dots, p_n$ in the plane determines a Voronoi diagram. This is a partition of the plane into cells $c_1, \dots, c_n$, one for each point. The cell $c_i$ covers the region which is closer to $p_i$ than to any of the other points of $P$.

Ten points in the plane and the corresponding Voronoi diagram (the cells on the border stretch out to infinity). The edge between $c_i$ and $c_j$ is part of the perpendicular bisector of the line segment (dashed line) joining $p_i$ to $p_j$. Image credit: Tony Phillips.

The points that determine the Voronoi diagrams in the Chinese money plant leaves are a set of secretory pores known as hydathodes.

Left: a photograph of a money-plant leaf marked up with yellow dots for the hydathodes and white lines for the vein system. Right: the Voronoi diagram (yellow) with the hydathodes as generating points, along with a copy of the actual vein system. Image credit: Fig. 1d–1e of Nature Communications 17 4111, lightly edited, used under a CC BY-NC-ND 4.0 license.

How does this fairly esoteric geometrical construction turn up in plant tissues? The authors credit the hormone auxin, which triggers vein formation in plants. Hydathode nodes are sources of auxin. When the feedback between plant tissue and auxin is strong, we get open branching patterns. In the case of weak feedback—such as that in P. peperomioides—the auxin from one hydathode interacts with auxin from neighboring hydathodes. The result is canals that divide the space into Voronoi cells. The authors illustrate this phenomenon with numerical simulations.

Three images showing five regions on a blue background. From left to right, the regions grow and press against one another, starting to resemble Voronoi cells.
Three numbered frames from a computer simulation of auxin-driven patterning. “Waves of high auxin concentration collide along ridges that bisect the space between the sources, producing a Voronoi diagram.” Image credit: Fig. 4E1–E3 of Nature Communications 17 4111, lightly edited, used under a CC BY-NC-ND 4.0 license.

This image, calculated from distributions of auxin concentration, can be compared with a purely geometric picture of the generation of a Voronoi diagram.

Colliding, uniformly dilating circles generate a Voronoi diagram. Image credit: Tony Phillips.

—Tony Phillips, Stony Brook University

Read other recent Tony’s Take columns.