Susie Mathre

Arrange six officers of different ranks and regiments in a square so that each line (both horizontal and vertical) contains a different combination of rank and regiment.  It seems simple enough, but this problem had some curious inconsistencies: why would it work for every number except two and six?  Two is obvious, but six sure isn’t.

That’s probably why Leonard Euler wasn’t able to conclusively solve the problem in 1779, and then it was over another century before Gaston Tarry proved that two and six are impossible in 1900. Another sixty years passed before three mathematicians were able to prove that any number except two and six are possible.

The Graeco-Latin square or Euler square is two Latin squares superimposed. These squares fall into the field of combinatorics, an area of mathematics with quite a long history starting with magic squares back in 0-1 BCE China. Magic squares use numbers (instead of Latin characters) and were often used to ward off evil spirits, cast out the devil, or on the other end of the spectrum, show reverence for gods. 

I enjoyed choosing the various seedhead and cone stand-ins including the bizarre pinecone galls that I found by chance while out skiing and waiting for a friend to catch up.  

I still find it challenging to find the two pairs of repeats in this piece - as there is no solution for an order six array, (unless you venture into the quantum route, but that’s another story or another piece of art).

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This is the third piece of three, affiliated with phyllotaxis and its connection to some very obscure math concepts that took a long time for me to work my head around.  

I came across an article titled, Growth in Plants: A Study in Number, by J. Kappraff, 2002. It sounded like just what I was interested in. Several weeks later, or maybe months, I was finally able to comprehend most of the concepts. It was bewildering to learn that phyllotaxis had links to Wythoff’s game, a Fibonnaci number system called Zeckendorf notation, the Rabbit Sequence, and Van Iterson’s Model. 

It was towards the end of the article that I came across an illustration of Adamson’s Primary Phyllotaxis Wheel, and I was drawn to try and understand it. After a long while it made sense, for the most part. “Plant phyllotaxis is an example of chaos in the spatial rather than the time dimension” was how the author put it, and that is a pretty good description as it turns out.

One of the most fulfilling parts of finally putting this together after weeks of cutting out the wheel, was choosing the colors and placement of waxed leaves I have collected over the years. All to illustrate the movement of petals as they grow, using plants in this puzzling, but captivating wheel of chaos.

This is the second piece of three, exploring Farey sequences, which are connected to Ford circles, which are connected to Fibonacci numbers and therefore phyllotaxis! 

Farey sequences are diagrams that show completely reduced fractions, (e.g. 2/4 can be reduced to 1/2), between 0 and 1. Ford circles are the geometric representations of the fractions, and the Fibonacci sequence numbers are found in the numerators and denominators of some Farey fractions. 

Ford circles are named after Lester R. Ford, Sr, who was an American mathematician who spent a year or so abroad teaching in Edinburgh where the student newspaper reported on Ford, “in amassing a collection of weird degrees, and a still more weird collection of mathematical lore, he is now busy shoveling it into the skulls of not too enthusiastic students”.

Aside from the fascinating history of tangent circles, I was drawn to this tree form of the Farey sequence. Wild rose hips seemed like they would make good vertices, if a bit prickly to handle, so it worked out well that I gathered them in winter. 

Having done several art pieces illustrating various aspects of phyllotaxis in plants, (the ordering and placement of leaves on a stem or petals on a flower) and the link to the Fibonacci sequence, I was looking for more mathematical connections. This is the first piece of three, loosely affiliated with phyllotaxy and Farey sequences.

Farey sequences are diagrams that show completely reduced fractions, (e.g. 2/4 can be reduced to 1/2), between 0 and 1. Fibonacci sequence numbers are found in the numerators and denominators of some Farey fractions. In researching the history of Farey sequences, I came across some unexpected findings.

In the 1747 edition of The Ladies Diary, an annual publication printed in London, that contained mathematical problems and other “Entertaining Particulars Peculiarly adapted for the Use and Diversion of the Fair-sex”, there was a math puzzle (from a man) that queried, “It is required to find (by a general theorem) the number of fractions of different values, each less than unity, so that the greatest denominator be less than 100?” This is equivalent to asking the size of the 99th Farey sequence. It wasn’t until the 1751 edition that they published three solutions, two of which were flawed. Turns out this problem is hard. It took the 18th century mathematicians 4 years to solve it. 

For myself, I enjoy the visual forms that Farey sequences can take, leaving the above question to be solved by someone else, perhaps of the Fair-sex. The configuration in this piece reminded me of searchlights in a night sky, in some fictional city from the Art Deco era of the 1920s. 

After the Friendship Paradox, I was on a (very short) roll of exploring other mathematical paradoxes and envisioning plants as stand-ins for the story. 

Zeno was a philosopher born around 450 BCE in Elea, Italy.  He was a student of Parmenides, a pre-Socratic philosopher.  Zeno was not a mathematician, and evidently died while attempting to assassinate the local tyrant of Elea.

There are ten paradoxes attributed to Zeno, and this one is one of the more well known, probably due to Aesop’s fable The Tortoise and the Hare, a similar story, but without a paradox, just a sleepy rabbit. 

Zeno’s arguments center on the idea that space and time are infinitely divisible, and was the first person to show that the concept of infinity is open to debate.  And debate they did, those early philosophers; it wasn’t until the invention of calculus in the late 17th century that we now understand why Achilles can catch the tortoise; the sum of an infinite series (where every term is greater than zero) can be finite. 

The winged seeds (samaras) of my Norway maple regularly helicopter down to the ground in my yard, which gave me the idea to collect various samaras of different maple species, which I then flung into the air to check their speed against one another.  It turns out that size matters, with the larger sugar maple quickly landing on the ground, while the dainty Rocky Mountain maple took its sweet time spinning far more revolutions and finally making its way to the earth.

When researching math puzzles, there are several paradox problems that have plagued mathematicians throughout history, and in this case social scientists as well.  It piqued my interest because it seemed to be improbable (or perhaps I didn’t want to address my level of popularity, or lack thereof).

Scott Feld really hit on something in 1991 when he first observed this paradox - that your friends have more friends than you do, and no one really wants to believe that it’s true.  But it’s been heavily studied and proven to be true whether we like it or not, and it has proven to be useful in the area of epidemics and their early detection.  When vaccines came out to help prevent the spread of the Covid virus, using the Friendship Paradox might have reduced the infection rate by a significant amount.

To illustrate this piece, I had to come up with a group of friends, and the characters from Alice in Wonderland came to mind.  I enjoyed revisiting the story and the characters.  I even purchased The Annotated Alice, a deep dive into everything about Wonderland, including interpretations behind what Lewis Carroll had been thinking when he wrote it and deciphered puns and riddles.  Who knew?

The choice of Flodman’s Thistle (a native thistle) seemed obvious once I started envisioning them as characters rather than prickly seedheads, and in the end, I’m okay with the concept that my friends have more friends than me.  I’m just grateful I have such wonderful friends.

I was attracted to this theorem because of its simplicity and beauty.  The idea that the intersection points of three pairs of tangent lines of three circles will all land on a straight line is rather lovely, and the proof is fairly clear-cut, not beyond all comprehension. Rather than explain the proof, the history of the man it is named for is perhaps more interesting. 

Gaspard Monge, born 1746 in the wine country of Burgundy France, proved to be gifted in many fields of study including mathematics, physics, geometry, and when he married into a family that owned a forge, metallurgy was added to the list. 

In league with Napolean during the French Revolution, Monge helped select art treasures in Italy to be brought back to France (now in the Louvre) to finance Napolean’s military campaign. During that time he also wrote a large number of scientific papers on math and physics, helped establish the metric system, and was made a count by Napolean only to have all his honors stripped after the fall of Napolean. 

Known today in the scientific community as the father of differential geometry, a statue of him was erected in 1849 in his home town of Beaune, and his name is one of the 72 names inscribed on the base of the Eiffel Tower. 

For this piece I spent a fair amount of time looking for round branches, something that doesn’t really exist in nature, so I had to make due with almost round. 

As G.K. Chesterton said, “Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait.”

What sort of person would come up with a puzzle of trying to draw a square and a circle of the same area using just a compass and straightedge? Someone with time on their hands.

Around 450 BCE, Greek mathematician Anaxagoras of Clazomenae was spending time in prison for claiming that the sun was not a god, but in fact a large luminous rock. What better place to ponder this famous math question of ‘squaring the circle’.

It has since been proven to be impossible, but not until 1882, because the area of a circle of radius 1 is pi, a transcendental number (meaning it’s not geometrically constructible with a compass and straightedge). But this hasn’t stopped people from trying...

Researching this showed that this is still a puzzle that can be solved, in spite of the pi problem. This piece has just three examples. Two are remarkably accurate and the third - not so much, but the shape of the castor bean leaf growing in my garden is what inspired me to investigate this puzzle.

One method I didn’t illustrate was the mystical vision of sacred geometry: a somewhat creative, if even less accurate method than my castor bean leaf.

If you throw out the compass/straightedge requirement then there have been several solutions, including one from 2022, when a group of three mathematicians figured out how to form a square with the same area as a circle by cutting them into interchangeable pieces that can be visualized. Never mind that their pieces number about 10 to the 200th.

While designing a new tiling pattern for my bathroom, I wondered about other options beyond the pedestrian square tile.  This is when I chanced upon tiling that never repeats itself, or aperiodic tiling.

Regular old periodic tiling has been around since early civilization and patterns that are repeating are ubiquitous all over the world, using shapes in beautiful and inventive combinations in some cultures, or in others, the banal bathroom...

This tryptych of pieces illustrates the three refinements that Roger Penrose made when discovering tiling shapes that would never repeat in the plane (aperiodic). Known as the Penrose tiles: P1, P2, and P3, they are based on 5-fold symmetry, or the pentagon. The challenging thing about pentagons is if you try to tile them, there’s always a gap left as a remainder.

This first piece shows the set of tiles that Penrose came up with based on the pentagon using four different shapes that comprise a star, a boat, a diamond, and three pentagons that fit together in a very particular pattern using rules about where each shape can be placed. The initial placement of these may appear to be in a repeatable pattern but the reality is that beyond this small grouping they do not repeat, illustrated with lines engraved in the paper.

Beyond the fact that Penrose thought pentagons “are just nice to look at”, they have some aspects that are found in nature: a regular pentagon uses the Golden Ratio (linked to the Fibonacci sequence) where the ratio of diagonals to the sides is 1.618 (Ф).

While designing a new tiling pattern for my bathroom, I wondered about other options beyond the pedestrian square tile. This is when I chanced upon tiling that never repeats itself, or aperiodic tiling.

Regular old periodic tiling has been around since early civilization and patterns that are repeating are ubiquitous all over the world, using shapes in beautiful and inventive combinations in some cultures, or in others, the banal bathroom... 

This tryptych of pieces illustrates the three refinements that Roger Penrose made when discovering tiling shapes that would never repeat in the plane (aperiodic). Known as the Penrose tiles: P1, P2, and P3, they are based on 5-fold symmetry, or the pentagon. The challenging thing about pentagons is if you try to tile them, there’s always a gap left as a remainder.  

This third piece shows an even simpler set of tiles that Penrose came up with based on the pentagon using two different shapes that use two different rhombi that fit together in a very particular pattern using rules about where each shape can be placed. The initial placement of these may appear to be in a repeatable pattern but the reality is that beyond this small grouping they do not repeat, illustrated with lines engraved in the paper.  

In 2023, a new single shape was discovered that could tile a plane aperiodically, known as ‘the hat,’ and is also called an einstein, which has nothing to do with Albert, but with the fact that 'ein Stein' is German for 'one stone'.

While designing a new tiling pattern for my bathroom, I wondered about other options beyond the pedestrian square tile. This is when I chanced upon tiling that never repeats itself, or aperiodic tiling.  

Regular old periodic tiling has been around since early civilization and patterns that are repeating are ubiquitous all over the world, using shapes in beautiful and inventive combinations in some cultures, or in others, the banal bathroom... 

This tryptych of pieces illustrates the three refinements that Roger Penrose made when discovering tiling shapes that would never repeat in the plane (aperiodic). Known as the Penrose tiles: P1, P2, and P3, they are based on 5-fold symmetry, or the pentagon. The challenging thing about pentagons is if you try to tile them, there’s always a gap left as a remainder.

This second piece shows a simpler set of tiles that Penrose came up with based on the pentagon using two different shapes that comprise a ‘kite’ and ‘dart’ that fit together in a very particular pattern using rules about where each shape can be placed. The initial placement of these may appear to be in a repeatable pattern but the reality is that beyond this small grouping they do not repeat, illustrated with lines engraved in the paper.  

In 1982, a form of aperiodic tiling, using ‘forbidden symmetry’ was discovered in the form of crystals, known as quasi crystals that occur naturally in nature. And this led to a new field of materials science which brought us, among other things, non-stick cooking pans. 

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”.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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