Math Art

Archimedes Dragon.  Archimedes is credited with making the first dissection puzzle using 14 polygonal shapes, (titled the Stomachion) . The aim was to construct a perfect square from the various polygons. The original was discovered in a palimpsest , written over by an anonymous medieval scribe compiling prayers. How many ways to 'solve' this was unknown until a computer scientist, Bill Cutler at Cornell University showed that there are 536 if you allow either side of the polygons to be used) . Below is an inset of Cuttler's array of the patterns.  Of course, Archimedes didn't have a fire breathing dragon in mind, but by adding the 'eye' to pattern F3 from Cutler's array , I thought it was a dead ringer. The frame is 1/4" thick copper with large brackets.

Pi-e.      Are all numbers the same?    They are not.   This painting depicts the mathematical constants pi and e, which are both transcendental numbers.  The digits of each are represented here as colors extending out to infinity. This sort of started out when Georg Cantor established that there were different levels of infinity .    Infininity is not the biggest number, rather it is a concept that is kind of hard to grasp for us humans who live in and experience a finite world .   The group of numbers we first learn are the  natural numbers you can count 1,2,3,4……………..on up to infinity . If we then include  the negative numbers - 1, -2, -3, -4  etc., we end up with the integer set of numbers.   Next, think of all the fractions you can  -  ½, 1/3/, 1/4, 1/5, 1/6, etc.  {In addition to being able to be written as a ratio, these numbers have the property of being able to be written  also as a decimal or a repeating decimal (for example, 4/11 = .363636363636..........  or 5/7 = .714285714285……..   So it would seem there are more of these  than there are whole numbers since there’s a more fractions in between 0 and 1 than just the one integer number increase to number 1.  Right?  Actually, not so, because  Cantor showed that for each fraction, you can pair it up with a whole number and count it.  For example:

½       1/3         1/4          1/5        1/6            1/7.................etc

 1         2           3            4           5            6  .............  all the way to infinity

Then we conclude that the integers and the rational numbers (fractions) both belong to the set of numbers which are countable, all the way to infinity.   All integers and rational numbers are algebraic  as are all roots of integers.  

Then mathematicians identified properties of a different set of numbers that are irrational - they cannot be expressed as a ratio ( a fraction).    An example is √2  (the square root of 2)  which goes on forever without repeating as  1.414121356……..  It turns out there are a LOT of these "real numbers' tucked in between the fractions. Cantor constructed an argument which showed that these numbers are uncountable and as such represent a set of numbers which is larger than the countable numbers – a second level of infinity if you will.   Also in this set are complex numbers – those which have a real and an “imaginary” component such as 4x + 3i ;         (i=  √-1     {i being defined as the square root of -1} )  Since some of the irrational numbers can be the root of an equation ( a famous example is phi, the golden ratio, which satisfies x^2 - x-1=0) ,  they qualify as an algebraic number, so not all irrational numbers are transcendental but all  transcendental numbers are all irrational and  are NOT algebraic - they do not solve a 'rooty' equation.  It has been shown that although most numbers are transcendental, it is quite difficult to prove that a given number is transcendental.  

for another piece of pi, look for Euler's pie below.  Sorry for the puns ............I can't resist.

Quadrature of the Loon (play on words).  In ancient times, one of the three unsolvable mathematical problems was to create  (using only a compass and straightedge - not a ruler!) a square with the same area as a circle,  which was termed quadrature . The problem still exists today. However, it is now known to be impossible because a circle's area is calculated using pi which is an irrational & transcendental  number). This problem originated with the Eqyptians taxed people based on their land ownership and so reduced areas to triangles or squares .  Circles and  most portions of circles (lunes) defied this method.   In Posey county, Indiana,  physician Edward J. Goodwin  published his solution  as "The quadrature of the circle" in 1897. He had drafted a bill for the Indiana House of Representatives to benefit the children of the state by showing his proof which had at its core redefining pi as 3.2 instead of 3.14159........etc.  stating that the claimed value with its infinite, laborious strings of decimal digits as wholly wanting and misleading.  Fortunately, the head of the math department at Purdue University had convinced politicians of the absurdity of this idea and the bill, which had passed in the House, was stalled in committee. of the Senate.   

So here, we have the solution - dividing a 'loon' (sic lune)  into quadrants.   -glass within a custom fabricated metal frame. (original)

Leonhard Euler.    This is a copy of an online portrait made in 1753 by his Swiss countryman, Jacob E. Handmann., which I made by using a metal punch and hammer on a panel of aluminum. 

Euler was one of the most eminent mathematicians of the 18th century and held to be one of the greatest of all time. He was also very prolific, publishing more than anybody in the field. His work in mathematics included  geometry, trigonometry, algebra, infinitesimal calculus, graph theory, topology and analytic number theory.  Euler is the only mathematician to have two numbers named after him. (e = 2.71828... and gamma , the EulerMascheroni constant=  0.57721.....  Important contributions include the concept of a 'function',  notation for trigonometric functions, Sigma for sums, and i to denote the imaginary unit.  He also popularized the use of the greek letter pi for the ratio of circumference to diameter.  His Euler diagrams were the forerunners of Venn diagrams.  He spent most of his life in St. Petersburg, Russia and also wrote works in mechanics, fluid dynamics, optics, astronomy and music theory. 

Tetrahedron & hyperbolic paraboloid  . These are just geometric shapes made in metal.  The tetrahedron is one of 5 regular, convex polyhedrons , constructed by congruent polygonal faces with the same number of faces meeting at each vertex.  Plato wrote about them in 360 BCE, linking 4 with the classical elements of earth, fire, water and air.  The tetrahedron having sharp points and like the heat of fire could produce stabbing pain.  The other regular solids were the cube with 6  square faces, the octahedron with 8  equilateral triangular faces, the dodecahedron with 12 pentagonal faces and the icosahedron with 20 triangular faces.  These classical solids were used by Kepler in the 16th century as a model for the structure of the known solar system of planets. 

The hyperbolic paraboloid is a doubly curved surface showing a convex curve along one axis and a concave on another axis. This results in every point being along the intersection of two straight lines, thus it is easily constructed and has therefore been used in modern architecture .  It also has the property of being efficient in materials for the strength it provides.     The next piece is another example of hyperbolic space.

Squared squares.   Squaring the square is a curious turn of phrase. It sounds as if it may have some relation to squaring the circle but it is so only in the sense that dividing a square into smaller different sized squares is a seemingly difficult task.   Although this would seem to be a problem of classical geometry it was not until 1902 that Henry , an Englishman who dabbled in puzzles and recreational math wrote about it. This figure of fused glass represents the smallest  number of different squares (21) which can made into a square itself with no gaps.  It was identified in 1978 by A.J.W. Duijvestijn.   It is a perfect solution because all the squares are different and no subset of them form a rectangle.  The smallest central square is 2 units and the largest in the left upper corner is 50.  The large square is 112 units on each side.  Fused glass. 

Srinivasa Ramanujan: 22 December 1887 – 26 April 1920)[2] was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation: "He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered".[3] Seeking mathematicians who could better understand his work, in 1913 he began a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Ramanujan had produced groundbreaking new theorems, including some that Hardy said had "defeated him and his colleagues completely", in addition to rediscovering recently proven but highly advanced results.

During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).[4] Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research.[5] Nearly all his claims have now been proven correct. (from Wikipedia).  Pastel on canvas. Although he was diagnosed with tuberculosis, medical authorities now  conclude that he died from hepatic amebiasis (a parasitic infection) at age 32. 

Georg Ferdinand Ludwig Philipp Cantor, (3 Mar 1845, St. Petersburg, Russia—6 Jan 1918, Halle, Germany), German mathematician who defined the cardinal and ordinal numbers and their arithmetic, founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another.  In 1873 Cantor proved the rational numbers  ℚ  countable, i.e. they may be placed in one-one correspondence with the natural numbers,  ℕ  . He also showed that the algebraic numbers, 𝔸,  i.e. the numbers which are roots of polynomial equations with integer coefficients, were countable. However his attempts to decide whether the real numbers ℝ were countable was more difficult  but in time he proved that the real numbers were not countable (i.e. are more numerous than the natural numbers) and his paper in 1874 implies that there is an infinity of infinities.  Cantor's work is of great philosophical interest, a fact he was well aware of. Personally, he was an accomplished violinist, a devout Lutheran (who thought his concepts were a gift from God) and suffered from bouts of depression.   Pastel on canvas.

Emmy Noether   

23 March 1882 – 14 April 1935) was a German mathematician who made important contributions to abstract algebra and theoretical physics.  She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics.  As one of the leading mathematicians of her time, she developed the theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws. She entered the University of Göttingen in 1903earned her PhD in 1907 and stayed on to produce excellent work in her field but only rose to assistant professor without tenure and a small salary. Her status did not change while she remained at Göttingen, owing not only to prejudices against women, but also because she was a Jew, a Social Democrat, and a pacifist.   In 1928-29 she was a visiting professor at the University of Moscow. In 1930, she taught at Frankfurt. The International Mathematical Congress in Zurich asked her to give a plenary lecture in 1932, and in the same year she was awarded the prestigious Ackermann-Teubner Memorial Prize in mathematics.  When the Nazis came to power in Germany in 1933, Noether and many other Jewish professors at Göttingen were dismissed. In October she left for the United States to become a visiting professor of mathematics at Bryn Mawr College and to lecture and conduct research at the Institute for Advanced Study in Princeton, New Jersey. She died suddenly of complications from an operation on an ovarian cyst. Pastel on canvas

Evariste Galois  

The story of ‘Evariste Galois is probably the most fantastic and tragic in the history of mathematics. A sensitive and prodigiously gifted your man, he was killed in a duel at the age of 20, ending a life which in its brief span had offered him nothing but tragedy and frustration.  Fortunately, for mathematics, the night before the duel he wrote down his main mathematical results and entrusted them to a friend.  When he was only a youth his father committed suicide, and Galois was left to fend for himself in the labyrinthine world of French university life and student politics.  He was twice refused admittance to the ‘Ecole Polytechnique, the most prestigious scientific establishment of its day, probably because his answers to the entrance examination were too original and unorthodox.  When he presented an early version of his important discoveries in algebra to the great academician Cauchy, this gentleman did not read the young student’s paper, but lost it.  Later, Galois gave his results to Fourier in the hope of winning the mathematics prize of the Academy of Sciences, but Fourier died and that paper, too, was lost.  Another paper submitted to Poisson was eventually returned be Poisson did not have the interest to read it through.  Galois finally gained admittance to the ‘Ecole Normale, another focal point of research in mathematics, but he was soon expelled for writing an essay which attacked the king.  He was jailed twice for political agitation in the student world of Paris.  In the midst of such a turbulent life, it is hard to believe that Galois found time to create his colossally original theories on algebra.

What Galois did was to tie in the problem of finding the roots of equations with new discoveries on groups of permutations.  He explained exactly which equations of degree 5 or higher have solutions of the traditional kind - and which others do not.  Along the way, he introduced some amazingly original and powerful concepts, which form the framework of much algebraic thinking to this day.  Although Galois did not work explicitly in axiomatic algebra (which was unknown in his day), the abstract notion of algebraic structure is clearly prefigured in his work.  This time his work was not lost, but it was only published 15 years after his death.  {from A Book of. Abstract Algebra by Charles C. Pinter, Dover Publ. 2nd ed. 1990. }  Info from : https://en.wikipedia.org/wiki/Galois_theory   

Pastel on canvas.

4 Color theorem .     This drawing appeared as a black & white cartoon in  Gary Larson's 1992 book , "A Farside Collection - Unnatural Selections" and I have taken the liberty  to color it (still with pencils)  in order to demonstrate the principle of the theorem .  The four color map theorem, first presented as a conjecture in 1852,  states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.[1] It was the first major theorem to be proved using a computer (1976). Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand.[2] Since then the proof has gained wide acceptance, although some doubters remain .  https://en.wikipedia.org/wiki/Four_color_theorem 

  Isaac Newton: (25 December 1642 – 20 March 1726/27[a]) was an English mathematician, physicist, astronomer, theologian, and author (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

In Principia, Newton formulated the laws of motion and universal gravitation that formed the dominant scientific viewpoint until it was superseded by the theory of relativity. Newton used his mathematical description of gravity to prove Kepler's laws of planetary motion, account for tides, the trajectories of comets, the precession of the equinoxes and other phenomena, eradicating doubt about the Solar System's heliocentricity. He demonstrated that the motion of objects on Earth and celestial bodies could be accounted for by the same principles. Newton's inference that the Earth is an oblate spheroid was later confirmed by the geodetic measurements of Maupertuis, La Condamine, and others, convincing most European scientists of the superiority of Newtonian mechanics over earlier systems.

Newton built the first practical reflecting telescope and developed a sophisticated theory of colour based on the observation that a prism separates white light into the colours of the visible spectrum. His work on light was collected in his highly influential book Opticks, published in 1704. He also formulated an empirical law of cooling, made the first theoretical calculation of the speed of sound, and introduced the notion of a Newtonian fluid. In addition to his work on calculus, as a mathematician Newton contributed to the study of power series, generalised the binomial theorem to non-integer exponents, developed a method for approximating the roots of a function, and classified most of the cubic plane curves.  Pastel on canvas

John Von Neumann  

28 December 1903 – 8 February 1957) was a Hungarian-American-Jewish mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields. Upon completion of von Neumann’s secondary schooling in 1921, his father discouraged him from pursuing a career in mathematics, fearing that there was not enough money in the field. As a compromise, von Neumann simultaneously studied chemistry and mathematics. He earned a degree in chemical engineering (1925) from the Swiss Federal Institute in Zürich and a doctorate in mathematics (1926) from the University of Budapest . Von Neumann commenced his intellectual career at a time when the influence of David Hilbert and his program of establishing axiomatic foundations for mathematics was at a peak.  Important work in set theory inaugurated a career that touched nearly every major branch of mathematics. Von Neumann’s gift for applied mathematics took his work in directions that influenced quantum theory, automata theory, economics, and defense planning. Von Neumann pioneered game theory and, along with Alan Turing and Claude Shannon, was one of the conceptual inventors of the stored-program digital computer.  Pastel on canvas.

   Liu Hui  劉徽    (225-295 CE) was a Chinese mathematician and writer who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art, in which he was possibly the first mathematician to discover, understand and use negative number. Liu Hui expressed all of his mathematical results in the form of decimal fractions (using metrological units).

Liu provided commentary on a mathematical proof of a theorem identical to the Pythagorean theorem.[5] Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".

In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry. For example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge.  He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. In his commentaries on the Nine Chapters, he presented an algorithm for calculation of pi (π) and worked it out to 3.141024 < π < 3.142074 {\displaystyle 3.141024<\pi <3.142074} a value between 3.141024 and 3.14107. He also devised a method to calculate the heights of various physical structures from a distance (i.e. trigonometry).

Tess (the pied horse) and friends  A tiling or tesselation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.  in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.  Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations.   MC Escher made use of natural topics  and often an evolutionary patterning to popularize  tilings. Quilting is a prime example of tiling.  Dr. Irina Swanson, Chair of the Dept of Mathematics at Purdue has demonstrated (on YouTube) several examples of how quilting makes use of geometric patterns.  Currently,  other mathematicians are using tessellations in higher dimensions and a variety of geometries. 

Also I've attached an homage to Escher of ' hand drawing hand'  just because I like it.  The Escher original is black & white (pen/ink) and so is mine (acrylic paint)! 

Desargue's Theorem 

Girard Desargue  ( 21 Feb 1591- Sep 1661 ) was a French mathematician and engineer who is considered one of the founders of projective geometry.  Projective geometry, as opposed to plain or Euclidean geometry, allows parallel lines to meet at infinity (contrary to Euclid's 5th postulate). The appearance of a distant scene is then  from a single point of perspective  (an example is  pictures of railroad tracks seem to meet at the horizon)    Although art in the 19th and 20th century often developed in response to advances in mathematical or scientific concepts, the perspective view in art arose as early as the middle ages and was further developed during the Italian Renaissance.    Projective geometry shares this concept and adds more elaborate purely  mathematical  concepts, mainly the idea that geometric characteristics remain unchanged by various mathematical transformations (such as rotations/reflections, inversions,  or more complex functions). These ideas, however,  did not develop until the 19th century.  They led to a much more complex  and imaginative understanding of space in general. Further advances in this field spawned  esoteric disciplines in mathematics including abstract algebra,  invariant theory, differential and finite geometry.  

Desargue's theorem states : denote the three vertices on one triangle by a, b and c, and those of the other by A, B and C.  Axial perspectivity means that lines ab and AB (blue lines in my painting)  meet in a point, lines ac and AC  (green lines) meet in a second point and lines bc and BC (red lines)  meet in a third point, and that these three points all lie on a common line called the axis of perspectivity.   The projected lines connecting the vertices of the triangles originate from the whitish circle (point) at the top center. This acrylic painting is an original of mine. 

Particle Wave - (23 x 41") ambigram after original by Douglas Hofstadter.  An ambigram is a construction of words/images that can be viewed in different ways. In this particular example, you may see the word 'particle' or the word 'wave' , either used to describe the basic nature of light.  Although this could reasonably be considered physics instead of math, I thought that it was too cool not to add to the collection. My wife actually did the mosaic, using dichroic glass and stained glass from Kokomo Opalescent Glass (which has been making glass in our town for over a century).  I only made the supporting structures.