Department of Mathematics

"Knight's Move": A mathematical art installation brought to you by the CMU Math Department and Studio Infinity

The sculpture
Knight's Move

What is this?

This sculpture is a mathematical graph (or rather, a visual representation of one), with "edges" made from aluminum strips connecting the "vertices" (joints) together. You'll notice that even though the vertices are flexible, the structure stands without collapsing (so far): this graph is what's called rigid. Rigid structures are an important part of bridges, skyscrapers, and the like, where engineers often incorporate triangles for structural stability due to their naturally rigid nature. Remarkably, however, this graph holds itself up despite containing no triangles at all! We can verify this by looking at the coloring of the edges: black and gold edges meet at some vertices, as well as red and gold, but no vertices connect both black and red edges together.

The graph's edges correspond to the possible moves of a knight piece from chess (2 spaces/vertices in one direction, and 1 in another), but on a 3-dimensional chessboard measuring 4 squares on each side. There are 288 such moves, which is enough to connect every space to every other. Interestingly, we can show mathematically that any large enough chessboard's squares are all accessible from any other square by enough moves of the knight (the graph is connected)

Finding rigid graphs with edges of a single unit length is an interesting mathematical challenge. It turns out versions of this graph (and more complex generalizations) exist for spaces of arbitrary dimension. For additional mathematical details and to view our source of inspiration for this project, please see this paper by Solymosi and White. Additionally, more details can be found here.

How was this built?

This structure was built by dozens of members of the CMU community throughout the day on August 29th, 2025; it is the result of a huge collaborative effort to assemble nearly 300 aluminum struts and hundreds of smaller pieces into a single, stable structure.

People building the sculpture People building the sculpture

Each anodized aluminum strut had to be carefully woven through the existing structure to ensure everything would fit. Joints between the struts are secured using ring-shaped connectors, demonstrating this shape's unique ability to hold itself up without rigid joints.

Close-up of the sculpture Close-up of the sculpture

The build took a total of 13 hours; a timelapse of the entire thing can be found here. We'd like to thank everyone who showed up to help (or just hang out), and especially to those who stayed late to complete the final touches!

Group photo of the builders

Links, Info, and Thank-You's

This project was also featured on the Mellon College of Science newsletter, feel free to check it out!

We'd like to extend our gratitude to Glen Whitney and the rest of Studio Infinity for the assistance in ideation, planning, and construction; to Prof. David Offner ("proffner" 😎) and the math department for their support and for funding the project; to the CMU Math Club for helping promote the event and convince people to show up; and anyone (once again) anyone who showed up on build day.

This project has been cleared by both Campus Design and EHS. It is temporary and will be taken down before the end of the Fall semester (<6 months). If you are looking for more information or have administrative concerns, please contact us by email (Tate Rowney, Allison Ma, Lou Feng, Dr. David Offner).

Website designed by TateRowney

Mathematically, an (undirected) graph is a set $V$ of objects called "vertices", along with a set $E$ of unordered pairs $\{v, w\}$ for $v, w \in V$ representing "edges" or connections between vertices $v$ and $w$. Often, additional conditions are put in place (for example, no edges such that $v=w$).
To make it more interesting, we've "embedded" this graph in 3D space ($\mathbb{R}^3$) so you can look at it. Embedding a graph $(V, E)$ in a space $S$ consists of finding an injective function $f: V \to S$ (representing which points the vertices are located at), and another injective function $g: E \to 2^S$ representing the paths our edges such that $g(\{v, w\})$ is homeomorphic to $[0, 1]$ (i.e. it's a line), and that $g(\{v, w\})$ starts at $v$, ends at $w$, and doesn't cross through any other vertices. There's usually some other requirements for $g$: for example, because we have to build it physically out of straight aluminum bars, the edges must also be relatively straight.
A graph $(V, E)$ embedded in $\mathbb{R}^n$ is said to be "infinitesimally rigid" if any possible movement of the graph's vertices involves only translating them by the same amount, i.e. their position relative to each other is unchanged. Mathematically, this holds if for any vector field $f: V \to \mathbb{R}^n$ such that $\forall (x, y) \in V$, $(f(x)-f(y)) \cdot (x-y) = 0$ (i.e. a tiny motion of the vertices), $f$ is equal to the restriction to $V$ of some other vector field $g : \mathbb{R}^n \to \mathbb{R}^n$ such that also $\forall (x, y) \in \mathbb{R}^n,$ $(g(x)-g(y)) \cdot (x-y) = 0$ (a tiny motion of the entire space).
In graph theory, we'd call these "3-cycles"; paths from a vertex back to itself that visit exactly three separate vertices.
A "path" between two vertices $v$ and $w$ is a sequence of vertices beginning at $v$ and ending at $w$ such that each pair of adjacent verticees in the sequence have an edge between them. A graph is "connected" if there is a path between any two vertices.
The length of any edge is of course relative to the embedding used (see above). A graph embedded in a metric space is called a "unit-distance graph" (sometimes called a "unit-bar framework") if the distance between any two vertices that share an edge is always the same.
trowney at andrew.cmu.edu
ama4 at andrew.cmu.edu
yufeng2 at andrew.cmu.edu
doffner at andrew.cmu.edu