Susie Mathre

This piece was started in winter, and finished during the following winter. The seeds of the Douglas fir are only visible on the ground where there is snow cover for them to contrast with. An easier way to collect the seeds would have been to harvest the cones when they were still closed, let them dry and later, when the cones open up, the seeds easily fall out. I learned this too late, but enjoyed collecting them during ski outings, where I could find them floating on top of the snow.

They are spotlighted in what’s known as an Apollonian gasket, named after Apollonius of Perga, a Greek geometer and astronomer from the early 2nd century BC who pondered about all things conic; i.e. circles, parabolas, ellipses. The gasket is a fractal generated from three circles tangent to the other two, successively surrounded with smaller and smaller circles, each maintaining tangency. The trick is to be able to find the size of the successive circle using the the radii of the surrounding three. It wasn’t until 1643 that French philosopher and mathematician René Descartes came up with a formula using the radii of four mutually touching circles, known as Descartes Circle Theorem.

The numbers in the gasket refer to the curvature, or bend, of a circle (defined as the reciprocal of the radius). The smaller the number, the larger the curve; hence a straight line has a curvature of zero. If the curvatures are denoted by a, b, c and d, then Descartes’ formula reads: a²+b²+c²+d²=(a+b+c+d)²/2. (The negative sign on the outer circle indicates that all other circles are internally tangent to that circle).

If math was not your favorite subject in school, the poem printed below the gasket delightfully describes Descartes Theorem in another form, written by English radiochemist Sir Frederick Soddy, who became enchanted by their charm in 1936. So much so, he published this poem in Nature, which he titled “The Kiss Precise”.

I was initially drawn to the visual beauty of Random Apollonian Networks (RAN) but then even more so to the terminology associated with them: arboricity, spanning trees, forests, tree decomposition, and their being something called “uniquely 4-colorable”. The research I did into these networks barely scratched the surface, mostly because it’s at a level of math far beyond what I’m familiar with or able to completely understand.

Generally speaking, a RAN is formed by subdividing a triangle into three smaller triangles, and then repeating the action of subdividing each newly created triangle into three more, and so on. As far as I can gather, these networks are useful to mathematicians who are interested in studying networks and how they might predict something like the way a disease will spread, who to vaccinate first, or how they can simulate what is known as “a small world effects phenomenon”, (sometimes called six degrees of separation), they can also simulate ways the internet works or how social networks function.

The aspect of RANs being 4-colorable is much easier to understand; it means that in a plane, such as a map or a planar graph (i.e. a Random Apollonian Network), you only need four different colors to fill in each region so that no area touches another with the same color. I used this four color theorem to fill in this network with leaves from trees in my yard, and some aspen leaves found on the trail, delineated in the diagram. The numbers on the vertices refer to the same diagram, and are “decomposed” to the right, showing the breakdown of each triangle.

I stumbled onto the Knight’s Tour while investigating some other facet of graph theory in Wikipedia, and fell down the rabbit hole (quite a ways this time), because the notion that there’s an entire subset of people who have spent time solving this puzzle is rather fascinating. I learned to play chess as a child, so I’m familiar with the game, but I had no awareness of the vast world of chess puzzles! The history of people trying to solve the Tour (the challenge being to find a path that the Knight travels, landing on each square only once) goes back to ancient times and is bizarrely compelling.

There are open tours and closed tours, tours on boards of different sizes and shapes, from smaller 5 x 5 boards, up to very large boards -130 x 130, to half boards. There are magical tours and semi-magical tours, and then there are types of tours; from Beverley tours to Irregular tours and several other types in between. The variations appear to be endless...

And because the Knight’s Tour may not be enough of a challenge, there’s the cryptotours for folks with a flair for combining poetry and tours around the board; albeit, a lot fewer of these seem to exist, and the craze for cryptotours seems to have crested during the late19th century.

Having previously collected a large amount of Houndstongue burs, I decided they would serve nicely as miniature horse heads making their way around a board covered with aspen and oak leaves. The path they take is also the answer to the cryptotour poem below it.

Thanks go to George Jelliss’ website, Mayhematics.com for his very thorough delving into the history and all things Knight’s Tour.

After completing pieces that focused on a circle, triangle, and a square, this diamond graph and its backstory caught my attention. This piece focuses on an practical query from 70 years ago that remains an unsolved math problem to this day.

During WW II, a Hungarian number theorist by the name of Pál Turán was forced to work in a brick factory near Budapest where his job was to push wheeled trucks full of bricks along rail tracks from kilns to storage yards. The trucks were prone to jumping the rails and spilling the bricks where the tracks crossed, causing loss of time and effort. This led Turán to wonder if there was a more efficient way to minimize the number of crossings.

After the war, Turán spoke about the brick factory problem while giving talks in Poland, and almost simultaneously in 1952, mathematicians Kazimierz Zarankiewicz and Kazimierz Urbanik both proposed solutions to the problem, with equivalent formulas for the number of crossings, achieving this by arranging the kilns along one axis and the storage yards along the other.

This is called the crossing number of a graph, which is defined as the minimum number of edge (line) crossings that occur when a graph is drawn in the plane (flat surface). I chose to use a double wire to represent the train tracks, with rose hips representing the color of bricks and the poppy seedheads representing the storage yards. The form these graphs take are examples of complete bipartite graphs. Several variations of these graphs and their crossings are illustrated in this piece using silk thread and bolts.

The mystery that remains is no one quite knows whether a graph can be designed whose layout contains fewer crossings than the arrangement shown here. Zarankiewicz’s conjecture was found to be erroneous 11 years after proposing it in 1952, and it still remains an open question today

This is an example of a dissection or transformation puzzle, something that I find fascinating. This is in the hinged category, and the challenge is to transform an equilateral triangle to a square in only four pieces. It was originally solved in 1903 by English author and mathematician Henry Ernest Dudeney, who later published a mathematical puzzle book with this dissection in 1907 titled, The Canterbury Puzzles and Other Curious Problems.

In a vigorous and verbose introduction (twelve pages), Dudeney states, among other notions, that, “A good puzzle should demand the exercise of our best wit and ingenuity” and, “They keep the brain alert, stimulate the imagination, and develop the reasoning faculties.” To that end, Dudney incorporated pilgrims from The Canterbury Tales by Geoffrey Chaucer (who are instructed to “beguile the ride by each telling a tale” in a story telling contest), as the characters who illustrate the math puzzles in Dudeney’s version.

I used four different grasses I collected in fall to weave a herringbone pattern suggestive of fabric and a haberdashery, and created the hinged pieces to be functional. I enjoyed the humorous narrative and Middle English used in the story, taking some liberties with the type sizes and layout of the story along with incorporating wood type I have in my collection.

This piece is a farcical attempt at alliteration involving a Venn diagram. Named for John Venn and introduced in 1880, Venn diagrams are useful in illustrating logical relations between finite collections of different sets, typically enclosed by circles or ellipses.

This venture began when I happened upon an article about the infinite earring, a geometric shape that illustrates topological space where the rings can be enlarged or shrunk infinitely within the defined space. This term, topology, led me to thinking about topography, and then typography, having spent many hours in mountainous terrain and many hours setting lead type.

Trying to then find a logical relationship connecting these three terms led me to only one plausible intersection: something related to religion, and blessedly it also began with the letter T. I can’t lay claim to any true understanding of typology, but then, I’m not entirely sure it’s meant to be understood, at least logically... It may be similar to a quote by John von Neumann, a mathematician and polymath, “In mathematics, you don’t understand things. You just get used to them.”

Putting this all together, I was happy to get the opportunity to set all my sizes of Centaur lead type, as a lover of topographical maps, it was fun to build a section of land that includes a hiking trail I’m fond of, and putting graphite into my compass from college days rekindled my love of using mechanical drawing tools

The Stomachion, or Archimedean Box, is two different ancient puzzles inside of a mystery. One puzzle is simply a game to create different shapes or animals with the pieces, the other is a math puzzle in combinatorics; exactly how many ways can the 14 pieces be rearranged to form a square?

The mystery encircles the journey of the treatises containing writings about the Stomachion that Archimedes wrote 2200 years ago, which were then erased and overwritten as a Byzantine prayerbook in 1229 AD. That prayerbook then disappeared, and finally resurfaced in 2000 as a project of the Walters Art Museum to reexamine the prayerbook (The Archimedes Palimpsest) in an effort to recover the treatises that lay hidden underneath the writings of the prayerbook on ancient parchment.

The Stomachion is of interest to me both visually and mathematically, and the mystery of Archimedes’ true intent in writing it is intriguing. The current theory is that his interest was in solving the combinatorial question - the number of ways it could be rearranged into a square (rather than just a puzzle to be arranged into other shapes). But it’s not an easy problem to solve, it took a team of four combinatorics experts six weeks in the early 2000s. Since then, programs have been written to conclude the final tally. Knowing this though, the mystery remains: how, and did, Archimedes ever solve it?

While creating this puzzle piece, I really enjoyed combining different leaves from local trees and perennials in my garden, and mounting them with beeswax onto individual hardboard panel pieces. Using silk thread on the lower part, I highlighted 14 of my favorite solutions to the Stomachion

This piece touches on one of the many applications of a Voronoi diagram, notably the ability to correlate sources of infections in epidemics. During the London cholera outbreak in 1854, British physician John Snow used a Voronoi diagram to illustrate that the proximity of people’s housing to a contaminated water pump correlated with contracting cholera.

The Covid pandemic was still in the early stages here in Montana when I was creating this piece, and I was wondering if there was any correlation between location and infection rates. After following the tracking data for a time, it became apparent that while there were higher numbers of cases where the population is more dense, there seemed to be a stronger correlation tied to income and travel, so this diagram is more a visual construct than a diagnostic one.

The Voronoi diagram initially caught my eye because the latticed network can resemble other patterns I’ve noticed in the natural world, especially the cell structure in leaves. Following this thought, each Voronoi cell in this piece (representing a county) has a section of leaf mounted on hard panel, then put together like a puzzle. Using another connection to nature and math, I decided to use numbers from the Fibonacci series to delineate the breakdown of cases.

This was a technical piece: it took numerous hours to hand saw, file, and sand the steel and plexiglass diagrams, which meant getting it right on the first try, but these kinds of challenges are very small in comparison to the challenges our planet faces as it heals from this pandemic. May we all recover soon and reinvent our future as a better place.

This piece is the first part of a triptych including the Salinon and Pappus Chain. All three figures are described in the Book of Lemmas, an ancient mathematical treatise attributed to Archimedes (c.287-c.212BCE) describing fifteen propositions (lemmas) on circles. The term "arbelos" translates as a shoemaker's knife in Greek, resembling the round blade on the knife used by cobblers.

I have an attraction to arcs, so I was drawn visually to the arbelos’ form and configuration; it can be constructed on a continuum, with the smaller two semicircles being any size as long as they are contained within the diameter of the larger semicircle. It’s a simple shape, but has many mathematical properties, a few of which I understand, and more that I have yet to study. The geometry that Archimedes was proving with the arbelos appeared straightforward, so I was especially drawn to the visual “proof without words”; it seemed possible to understand if one has a bit of background with the Pythagorean theorem.

I collected a large number of aspen leaves during autumn, and I love the subtle variations in their colors, so I took advantage of the relatively large area of the arbelos to combine the elegance of this geometric shape with the pattern of fallen leaves.

This piece is the second part of a triptych including the Arbelos and Pappus Chain. All three figures are described in the Book of Lemmas, an ancient mathematical treatise attributed to Archimedes (c.287-c.212BCE) describing fifteen propositions (lemmas) on circles. The term "salinon" is Greek for salt cellar, with the lower curve (supposedly) resembling an ordinary type of salt cellar from much earlier times.

The figure of the salinon is appealing to me, it’s graceful and combines four arcs. I like that there’s fluidity in how it can be drawn: the lowest semicircle can be any size as long as it is centered between the outer two semicircles and contained within the diameter of the top semicircle. The visual “proof without words” for the salinon wasn’t as obvious for me to understand as that of the arbelos, I ended up needing help from someone more familiar with some of the geometry concepts.

One of my neighbors allowed me to collect some Ohio Buckeye leaves from their tree to use for this piece. I was attracted to the dramatic angle of their veins, choosing to take care in their placement (in contrast to the arbelos), combining the flowing shape of the salinon with the dynamic aesthetic of the leaves.

This piece is the third part of a triptych including the Arbelos and the Salinon. All three figures are described in the Book of Lemmas, an ancient mathematical treatise attributed to Archimedes (c.287-c.212BCE) describing fifteen propositions (lemmas) on circles. The Pappus chain figure is mentioned at the end of Proposition 6 and named for Pappus of Alexandria, who investigated the figure in the 3rd century AD.

Pappus’ proof, which relied on Euclidean geometry, is quite long, so I chose to learn about the modern proof which is shorter and uses a rather fascinating method called circle inversion (attributed to the Swiss mathematician Jakob Steiner in the 1820s). It’s a concept that once I understood it, it all made sense, but I found it challenging to grasp at first. After I finally gained some level of comprehension, I wanted to illustrate the circle inversion of a Pappus chain, mostly for its pleasing appearance, forgoing any real attempt at a proof without words.

The visual beauty of the Pappus chain is just as engaging to me as the mathematical aspect; so I showcased each circle using leaves with vibrant colors. I always enjoy a good challenge with hand sawing, and then filing smooth the steel. I’m happy when I get it right in one go, especially if it requires several hours/days. I used silver rod for the rivets, stacking three sheets of steel: the top sheet to frame the circles holding the leaves and wax, and the lower two providing thickness for the bolts attaching it to the panel to be threaded from the back.