Cavalcade – Mathematical Connections

The diagrams on this page sketch out a modernization of the Archimedes calculation of the volume of a sphere. This achievement is particularly impressive as Archimedes derived the volume two millennia before the advent of calculus, at a time and place where zero and negative numbers were not accepted concepts.

The essential argument is that the volume of the sphere plus the volume of the cone must equal the volume of the cylinder because the corresponding relationship holds for the area of their two-dimensional cross sections at each height. Given that the volume of the cylinder is πr²h = 2πr³, and the volume of the double cone (a simpler formula to derive) is 2(1/3)πr²h = (2/3)πr³, we obtain the modern formula for the volume of a sphere, (4/3)πr³.

This image is an example of an Algebraic Starscape, described by Edmund Harriss, Kate Stange and Steve Trettel in the paper Algebraic Number Starscapes. These intricate patterns show the beauty of solving polynomial equations. The dots in this image represent the complex roots of cubic polynomials ax³+bx²+cx+b = 0, with the size getting smaller as the polynomial gets more complex (naively this is related to a,b and c getting larger, but is actually the cube root of the discriminant). The point the whole image seems to be pulling towards is i, the square root of -1.

The dots in this image represent the complex roots of cubic polynomials ax³+bx²+cx+b = 0, with the size getting smaller as the polynomial gets more complex (naively this is related to a,b and c getting larger, but is actually the cube root of the discriminant). The point the whole image seems to be pulling towards is i, the square root of -1.

In 1924, George Pólya published a paper in Zeitschrift für Kristallographie in which he proved that there are exactly seventeen wallpaper groups. In other words, if you look at designs in the plane that repeat in two non-parallel directions, like those on the side wall of the bakery and on the back wall of the curio shop, they all have one of seventeen different symmetry structures. This figure from his paper shows a representative picture for each wallpaper group.

At the time, Pólya was blissfully unaware that Evgraf Federov had already proven this theorem 33 years earlier. Nonetheless, the 1924 paper had a lasting impact on mathematical culture. Early in his artistic career, M.C. Escher came upon Pólya’s paper and his classification diagram, which overlapped Escher’s own explorations of regular tilings of the plane. As documented by mathematician and Escher biographer Doris Schattschneider, Escher copied each of the Pólya tilings into his notebooks, studied them carefully, and shared his admiration and gratitude in the letters he exchanged with Pólya.

This page is from Gauss’s mathematical diary; we can see his entry of what became known as Gauss’s Eureka Theorem. The theorem states that every positive integer can be expressed as the sum of three triangular numbers. A number is triangular if it counts the number of dots from a triangular lattice that fall within an equilateral triangle. The larger the triangle, the larger the triangular number; the first 6 triangular numbers are 0,1,3,6,10,15.

On the sheet one can read, in Gauss’s handwriting,

EYPHKA: num = Δ + Δ + Δ

The picture on this sheet illustrates several aspects of the hyperbolic plane in one combined visualization. From left to right, it shows (part of) a tessellation of the Poincaré disk by regular (hyperbolic) triangles, then a further subdivision into smaller hyperbolic triangles of the triangulation in which some of these smaller triangles are colored yellow to generate a pretty pattern.

Near the right, this triangulation “lifts off” into a 3d visualization that shows all the small triangles as equal in Euclidean size, requiring lots of “flounces” in the surface to provide enough area to accommodate them all – this is reminiscent of the crochet models of hyperbolic geometry in the Garden and the Reef.

The Farey sequence of order N is the linearly ordered collection of all fractions of type p/q, in which p and q are mutually prime positive integers, with p between 1 and q-1, and q not exceeding N. The Farey sequences have surprisingly sophisticated mathematical properties for such mundane objects. The figure on the sheet illustrates relationships between low-order Farey sequences and circles in the Apollonian gasket sandwiched between the horizontal axis and the two circles with radius ½ and centers at (0, ½) and (1, ½) respectively.

A dihedral group is the group of symmetries of a regular n-gon. In other words, it is a system of arithmetic built from the 2n different ways of rotating and reflecting the n‑gon; in this system, we can combine pairs of motions just as we can, for instance, add pairs of numbers.

This sheet (or rather collection of small sheets) shows concretely the action of the symmetries of the dihedral group of the square, called D₄ by some (geometers, because it is constituted by the symmetries of the 4‑gon) or D₈ by others (algebrists, because the group has 8 elements). The Mathemalchemy mouse from the mats around the Curio Shop undergoes reflections and rotations galore, each indicated by its own color. Pure rotations are various shades of pink/red; a reflection adds some blue. The big colored table shows the multiplication table (or Cayley table) of the group; other figures show the subgroup structure of this dihedral group.

The Fano plane is the smallest finite projective plane; it has only 7 points. In a projective plane every two points determine a unique line going through both points, and every two lines intersect in a unique point. The Fano plane has 3 points on each of its 7 lines, and 3 lines going through each of its 7 points.

To make a drawing of this on a Euclidean plane, some of the straight lines have to be drawn definitely non-straight. The figure on the right shows off the symmetries better, at the cost of not having any line looking straight. The figure on the left also functions as a mnemonic for the multiplication table of the octonions.

This Venn diagram illustrates connections between math, art, and abstraction; the imagery associated with each region corresponds to the region’s characterization as a set. The Math-only region features secant and tangent lines (key concepts in differential calculus), the intersection of just the Math and Abstraction circles contains a commutative diagram, and the intersection of all three circles contains an adaptation of Coxeter’s 30-45-90 tiling of the hyperbolic plane, an abstract design that appeals to mathematicians and artists. The Math and Art region features an Escher-like tiling by fish patterns, created by Bronna Butler; some of the fish swim into the Art-only region, and then escape the Venn diagram altogether. The title refers to how the colors in the circles’ intersections are created from the colors in the single-set regions. There are different ways in which colors can be mixed; “additive mixing” refers to the process of mixing colors using two or more differently colored light beams. Red, blue, and green fill the single-subject regions; their pairwise additive mixtures magenta, yellow and cyan fill the 2-region intersections; and the additive mixture of all three original colors forms the white in the center of the diagram . This piece was selected for the Mathematical Art Gallery at the 2021 Joint Mathematics Meeting, and you can watch the following video about it on Vimeo.

This sheet relates to the squirrels in the garden exploring primes with the Sieve of Eratosthanes. As they find the primes, the squirrels may notice that certain columns of their table have more primes while others have fewer. They might ask how the primes are distributed into these columns, which amounts to asking: how many primes have, in our standard bas-10 notation, a last digit equal to one of the 10 possible values, 0, 1, 2, .., 9?

Of course, certain end digits, like 4, cannot occur in any prime. Any number that ends in 4, 6 or 8 is divisible by 2 and therefore not prime. In addition, there is only one prime that ends in 2 (namely 2 itself), likewise there is only one prime that ends in 5 (namely 5 itself). So the squirrels might really be asking: how many primes ending in 1 or in 3, 7, 9? Are there more of some than the others?

The answer to this question is somewhat surprising and mysterious. On the one hand, Dirichlet’s Theorem on primes in arithmetic progressions says that in the long run (that is, as we take the limit to infinity of the sizes of primes we’re considering), the primes are evenly distributed between these four possibilities. On the other hand, if we stop at any finite point, it seems there are more primes ending in 3 or 7 than primes that ends in 1 or 9! The sheet shows some data illustrating this phenomenon. Even though it has been proved that Team 3‑and‑7 holds the lead for much of the time, Team 1‑and‑9 does take the lead infinitely often. Exactly understanding this race (and other similar races) is still an area of active research.

When drawing a diagram on a 2-dimensional sheet of paper, for 3 sets all contained in a larger set S, in which each of the 3 sets is represented by a circle and the sheet itself represents S, it is easy to arrange the circles so that the diagram shows all eight possibilities for an element of S (it could belong to none of the three smaller sets; there are 3 ways in which it could belong to one smaller set but not to the two others; there are again 3 ways in which it belongs to two smaller sets but not the third; finally it could belong to all three). This is called a Venn diagram; for 3 sets it is moreover easy to arrange the circles symmetrically.

Already for 4 sets it is not possible to draw a Venn diagram (which would now show 2⁴ = 16 regions) where the boundary of each of the 4 sets would be a circle: one needs to consider other shapes, and replace the circles by more general Jordan curves.

If one imposes that each of the 2⁴ regions is connected, then a symmetric arrangement is not possible: the four Jordan curves cannot simply be the same curve in four versions, obtained by rotating the same template for all four. David Henderson’s theorem states that a Venn diagram for N sets in which all 2^N possibilities are represented by connected regions, with each of the N sets delimited by a Jordan curve, can have rotational symmetry if and only if N is prime. The sheet shows some of the preliminary drawings made by David Henderson when he was working on this, and some drawings of the Venn diagram with rotational symmetry for small prime N. This paper contains a nice discussion and also points out and corrects a shortcoming in the original proof.

This sheet illustrates a geometric way of viewing and extending Dehn’s Lemma. Dehn’s lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map’s singularity set in the disk’s interior, implies the existence of another piecewise-linear map of the disk which is an embedding and is identical to the original on the boundary of the disk. The proof of this theorem has a curious history: it was thought to be proven by Max Dehn in 1910, until Hellmuth Kneser found a gap in the proof in 1929; its status then remained in doubt until Christos Papakyriakopoulos proved a generalization in 1957, now called the loop theorem. This result was immensely important in the development of 3-space topology. A further extension in 1965, in David Henderson’s PhD thesis, resulted from his more geometrical interpretation, in which the question was rephrased as taking a given singular disk whose interior does not intersect the boundary, “change” it into a non-singular disk which has certain desired properties in common with the original singular disk..

The same argument as used in the traditional geometric proof of Pythagoras’s theorem [ (1) dropping perpendiculars from each vertex of a triangle onto the opposite side, and continuing it into the square built on the side, and then (2) showing equality of areas of rectangles by showing congruence of triangles that have exactly half the area of each of those rectangles], can be used for arbitrary (instead of straight) triangles and leads to an interesting observation extending Pythagoras’s theorem and making the argument more symmetric. Certainly something for triangles-turned-butterflies to celebrate!

Fermat proved this beautiful fact: that any prime that is 1 mod 4 can be written as the sum of two squares and conversely, if an odd prime can be written as the sum of two squares, it must be 1 mod 4. Here, we see a different proof of this fact that uses Minkowski’s Theorem, which states that given a lattice, any convex region that is symmetric about the origin and has sufficient area must contain a lattice point besides the origin.

Tetrahedral kites were first proposed by Alexander Graham Bell (maybe more famous for his pioneering work on the telephone); the sheet shows the header of his article on the subject. This type of kite has wind-catching sails at several angles on a very stable skeleton—the regularity of the tetrahedron leads to a strong shape with good load balancing.

Bell moved quickly from a single tetrahedron model to one with several cells, and even early ones already showed a “fractal-type” design reminiscent of the 2-dimensional Sierpiński triangle. This triangle is also “hidden” in Pascal’s triangle—just mark the locations of the odd numbers!

Emmy Noether was, by all accounts, an astounding mathematician as well as a fun person—her own favorite picture of herself was one in which she is on a boat, laughing at the photographer.

The sketch here was made by Stephanie Magdziak, in preparation for Stephanie’s creation of the bronze commemorative plaquettes that are now given to the ICM Emmy Noether lecturers at the quadrennial International Conference of Mathematicians. The two formulas were typeset for that plaquette as well. (More information can be found in this write-up about those plaquettes(PDF – 2.3MB).

The formulas refer to the two results for which Emmy Noether is most known: the formulation of the “ascending chain of principal ideals” condition, a fundamental property of special rings, now called Noetherian rings, and the Noether Theorem, stating that every invariance of a physical system under a group of transformations is linked to a conservation law, a basic result in mathematical physics. The printed page is the start of the paper on that second result. Amazingly, these two foundational results are basic building blocks in two mathematical subdisciplines now so far apart that their practitioners often don’t even know that Emmy Noether is also celebrated by members in the other discipline.

Every knot has a corresponding knot complement, which means if you take S³=R³ U {∞} and remove the knot (which is an embedded circle), the resulting space is called a 3-dimensional cusped manifold. The cusp is precisely where the knot was removed. Corresponding to each knot complement is a polyhedral decomposition, a way to describe the geometry of the manifold. The knot in this sheet illustrates the Figure-Eight knot and its corresponding decomposition into two ideal (vertices removed) tetrahedra. The arrows and colors illustrate how the two tetrahedra should be glued together in order to obtain the Figure-Eight complement.

The Figure-Eight knot has the smallest hyperbolic volume. This decomposition was first demonstrated by William Thurston in his notes The Geometry and Topology of Three Manifolds.

Wavelets are building blocks for wavelet transforms, in which more general functions are decomposed into a linear combination of scaled and translated versions of the template, the wavelet. Such transforms are useful in settings where many scales are in play. For instance, wavelet transforms are used in image processing and in the understanding and description of singularities in differential equations or integral operators.

For some specially constructed wavelets, the scaled and translated versions used in the wavelet transform constitute an orthonormal basis; they are linked to transform algorithms with very fast numerical implementations that use convolutions with short digital sequences (also called filters). The surface (of which the sheet shows two views, from the “front” and “back”) illustrates a 1-parameter family of such special basis-generating wavelets corresponding to digital filters with just 4 coefficients, going from the Haar wavelet to the “ferocious wavelet” painted by OctoPi; D4 is about 2/3 along the way.

Election of representatives to Congress in the US is organized by state; the number of House representatives for a state is (roughly) proportional to its population. States with more than one House representative are divided into congressional districts that each elect one representative. The boundaries of these districts can be redrawn every 10 years, to ensure (approximate) equal populations per district. District boundaries can also take other factors into account; the authorities redrawing boundaries are sometimes accused of “unfair” gerrymandering.

Mathematicians have developed nonpartisan, algorithmic tools to evaluate the “fairness” of a district map, for instance by comparing its electoral outcome to the outcome distribution for maps that are geometrically similar. Pictures on this sheet are borrowed from several such studies, coordinated by Moon Duchin and by Jonathan Mattingly.

Patterns occur naturally in nature. The shells of mollusks such as the Pearly Nautilus, Syrinx aruanus and Tectus niloticus (synonym: Trochus niloticus) have an elegant spiral structure that follows an “equiangular spiral” also known as a “logarithmic spiral’. For any rotation angle, the distance from the origin of the spiral increases by a fixed amount.

More information here

This sheet displays the first page of one of the NASA technical reports by Katherine Johnson, whose hand calculations were essential for many of NASA’s first manned space flights in the 1950s and 1960s. In the years leading up to this work, she was already a mathematical pioneer, recruited from her job as a public school teacher to be one of the first three black graduate students at West Virginia University. Her most celebrated contributions to the US space program were her calculations for John Glenn’s orbital flight in 1962. Because of the complexity of the flight path, NASA had set up a new network of computers and tracking stations to enable the mission, but the machines were prone to glitches and the astronauts were reluctant to trust them. Glenn himself famously refused to undertake the mission until Johnson had checked each of the computer results by hand.

Katherine Johnson is one of the African-American women mathematicians and engineers featured in the 2016 book Hidden Figures (written by Margot Lee Shetterly) and its film adaptation, a long overdue tribute to their historic achievements. In the previous year, at the age of 97, Johnson received the Presidential Medal of Freedom in recognition of her groundbreaking work in space exploration.

From early education we’ve been ingrained with the notion that the sum of the angles in a triangle always equals 180 degrees, i.e. π radians. But this is only part of the story. This story dates back to approximately 2300 years when Euclid stated the five geometry axioms. The fifth one known as the parallel postulate states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

More popularly, the parallel postulate is equivalent to the following statement: through a point not on a given line there is exactly one line parallel to the given line. This statement can be shown to imply that the sum of the angles in a triangle must equal π radians.

For two millennia, mathematicians tried to prove the fifth postulate from the previous four, to no avail. In the 19th century, mathematicians such as Lobachevski and Bolyai discovered a new geometry by selecting an alternative fifth axiom, in which one assumes that through a point not on a given line there are at least two lines parallel to the given line. This results in a geometry where the sum of angles in a triangle must be less than π radians.

One can also consider other alternatives to Euclid’s fifth axiom, and build a non-euclidean geometry that way. More precisely, one could assume that through a point not on a given line there are no lines parallel to the given line. An example of such a geometry is spherical geometry, with great circles taking on the role of straight lines. For triangles on a sphere the sum of the three angles is always more than π radians.

Three sibling sheets show triangular figures for the three geometries. In the hyperbolic case, the sum of the angles is less than π radians; in the elliptic case, the sum exceeds π radians. In both cases, the value of the difference equals the area of the triangle. The euclidean case, which separates the elliptic from the hyperbolic and can be viewed as the limit as the radius of the sphere (pseudosphere) approaches infinity, the sum of the three angles equals exactly π radians for all triangles, and gives no information about their area.

Famous mathematician Vladimir Arnold illustrated the mixing properties of a simple map from the square [0,1]² to itself by drawing a cat on the square, and showing how the black-and-white sketch was transformed by the map. This picture and construction have become known as “Arnold’s cat”; they inspired the name of the Mathemalchemy baker. The map as illustrated here consists of several steps: first, the linear transformation of R ² with matrix [1 1;1 2], which maps [0,1]² to a parallelogram; next, the pieces sticking out of [0,1]² are moved back into [0,1]² by adding the appropriate integer multiples of the vectors [1;0] and [0;1] – sections that require a different transport vector are given a different color. The four resulting triangles nicely tile [0,1]². As a result of the operation, the cat has been compressed in one direction, and “smeared out” in another. Repeating the map over and over again will see the transformed cat image approach a constant uniform gray.

The Baker’s map is another map from [0,1]² to itself that is strongly mixing. In the traditional Baker’s map (according to mathematicians), the square is first “rolled flat” (by applying the linear transformation with matrix [2 0;0 ½]) and then the piece sticking out into the neighboring square is “cut off” and placed “back on top” by translating it by the vector [-1;½]. However, true bakers are more likely to fold their rolled-out dough — which is why we show a culinarily more faithful version, with a cat “folded over”; this map is also strongly mixing.