Tuesday 23 July 2024

On This Day in Math - July 23


Men love to wonder, and that is the seed of science.

-Ralph Waldo Emerson

The 204th Day of the Year
204 is the eighth tetrahedral number, the sum of the squares from 1 to 8. Answers the question, how many squares are there on an 8x8 checkerboard.

204 = 20^2 - 14^2=52^2 - 50^2

204 is the sum of consecutive primes in two different ways: as the sum of a twin prime (101 + 103) and as the sum of six consecutive primes (23 + 29 + 31 + 37 + 41 + 43). (one might wonder what is the smallest number that is the sum of consecutive primes in more than one way... And what is the smallest prime number that is expressible as the sum of consecutive Primes in more than one way?)

And a trio from *Derek Orr @MathYearRound :
204 = 1²+2²+3²+4²+5²+6²+7²+8².

Sum of first 204 primes is prime.

100...00099...999 (204 0's and 204 9's) is prime.

204 is a refactorable number, it is divisible by the count of its divisors, 12

It is because it is divisible by 12 that 204 is expressible as (n+3)^2 - (n-3)^2 for n= 17 (204/12=17)

204^2 = 41616, a number that is both a square and a triangular number, the 288th triangular number. They are pretty rare as it is only the fourth. And like 204, it is the sum of twin primes also, 41616 = 20807 + 20809. Only 12 and 84 also are the sum of twin primes with a square that is also the sum of twin primes.
since I've written twin primes so many times in this entry, its probably a good time to remind you that, although the idea of primes separated by only a single composite number was known back to antiquity, the term was created around the end of the 19th century by German Mathematician Paul Stackel (in the German, of course, "Primezahlzwilling")

On an infinite chess board, a Knight can reach 204 different squares in eight moves.

200 is CC in Roman numerals, and 204 is CC in hexadecimal.

For More Math Facts for Every Year Date


594 The Sun was well up (17°) at 6:11 am when totality occurred. On a warm summer's morning it must have got surprisingly cold as totality approached, giving a clue that something unusual was about to happen. At 258 km wide this was an Eclipse with a very wide track and a good duration of over 3 minutes. The Eclipse track traveled into Denmark, Norway, Sweden, Finland, Estonia and Russia. *NSEC

1754 Joseph Louis Lagrange, 18, published his first work in the form of a letter in Italian (He was Italian born. Only his great-great-grandfather Lagrange was French, all other ancestors were Italian). A month later he realized that he had rediscovered Leibniz’s formula for the n-th derivative of a product. *VFR

1788 Jefferson's interest in surveying, and measurement in general led him to inquire of Benjamin Vaughan, then of England, about a British odometer, "I have heard that they make in London an Odometer, which may be made fast between two spokes of any wheel, and will indicate the revolutions of the wheel by means of a pendulum which always keeps it’s vertical position while the wheel is turning round and round. Thus [see Fig. 1.] I will thank you to inform me whether it’s indications can be depended on, and how much the instrument costs. " *Jefferson Letters

1829, William Austin Burt, a surveyor, of Mount Vernon, Michigan, received a patent for his typographer, a forerunner of the typewriter (U.S. No. 5581X). The Patent Office fire of 1836 destroyed the original patent model. Burt's typographer was a heavy, box-like contraption, made almost entirely of wood. Like today's familiar toy typewriter, the typographer had type mounted on a metal wheel, with a rotating, semicircular frame. By turning a crank, Burt was able to move the wheel until it came to the letter he wanted. Then he would pull a lever, driving the type against the paper and making an inked impression. *TIS

 It was the first typewriting machine to be patented in the United States, although Pellegrino Turri had made one in Italy in 1808.Perhaps because of its slow speed, or because there was not yet a wide market for typewriters, it was not a commercial success

The working model that Burt constructed for his 1829 patent was destroyed in the 1836 Patent Office fire


1904 The ice cream cone was introduced at the St. Louis world’s fair.*VFR by some accounts, the ice cream cone was invented by Charles E. Menches during the Louisiana Purchase Exposition in St. Louis. *TIS

To Whichever or both, Well done.  (That is an attempt at humor, the two names are both used for the same event.)

Although Menches is often named as the first to serve his ice cream in a rolled waffle cone, its true originator is far from clear. The edible cone was widely sold at the fair, and various other people there are also claimed as the innovator. *Tis

1927 The English term Eigenvalue first appears in a letter to Nature from A. S. Eddington beginning “Among those ... trying to acquire a general acquaintance with Schrödinger's wave mechanics there must be many who find their mathematical equipment insufficient to follow his first great problem—to determine the eigenvalues and eigenfunctions for the hydrogen atom" *Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics
The term was derived from Hilbert’s use of “Eigenwert” in 1904, although Helmholtz had earlier used “Eigentöne” for what he called proper tones in acoustics. proper was also used by many English and American scientists throughout the 1920-1950 period. Even J. Von Neumann’s translations into English used the term “proper“.
Paul Halmos followed von Neumann’s English writings and used “proper value” in his widely-used Finite Dimensional Vector Spaces (1958). However Halmos admitted defeat in his A Hilbert Space Problem Book (1967, p. x):

“For many years I have battled for proper values, and against the one and a half times translated German-English hybrid that is often used to refer to them. I have now become convinced that the war is over, and eigenvalues have won it; in this book I use them.”


1962 Tens of millions of people watched a historic broadcast as Telstar beamed live transatlantic video into viewers’ living rooms for the first time. The age of satellite television had dawned.
In homes across Rome, people barely touched their dinners. London’s pubs were packed, but bartenders served nary a drink. Throughout Europe, more than 100 million people huddled around television sets on the evening of July 23, 1962, to tune in to history. With Europeans watching eagerly, a black-and-white image of the Statue of Liberty flickered onto their screens. The picture itself was not particularly noteworthy except for one thing: it was live, via satellite. *History.Com
The first broadcast was to have been remarks by President John F. Kennedy, but the signal was acquired before the president was ready, so the lead-in time was filled with a short segment of a televised game between the Philadelphia Phillies and the Chicago Cubs at Wrigley Field before the President's address. A routine fly ball hit by Phillies second baseman Tony Taylor made telecommunications history.I have a different date and perspective from a BBC History page.  "On 11 July 1962 British television viewers saw pictures beamed live from the US via the Telstar satellite. Raymond Baxter and Richard Dimbleby were on hand to provide commentary, although the precise time of the broadcast was not known in advance. The first pictures, received in Britain just after 1am, were of the chairman of AT&T, Frederick Kappel, and of poor quality, while in France they were picked up clearly. However this landmark transmission marked the beginning of satellite broadcasting, and changed the face of telecommunications."

Telstar 1 was the first satellite capable of relaying television signals from Europe to North America. The 171-pound, 34.5-inch sphere loaded with transistors and covered with solar panels was placed in orbit by a Delta rocket launched from Cape Canaveral on July 10, 1962.

The maiden program included more images from around America, such as the Mormon Tabernacle Choir singing at Mount Rushmore and President John F. Kennedy conducting a press conference. 

1985 the legendary Commodore Amiga was released In 1985 Commodore revolutionized the home computer market by introducing the high end Commodore Amiga with a graphic power that was unheard of by that time in this market segment. Based on the Motorola 68000 microprocessor series the Amiga was most successful as a home computer, with a wide range of games and creative software, although early Commodore advertisements attempted to cast the computer as an all-purpose business machine. In addition, it was also a less expensive alternative to the Apple Macintosh and IBM-PC as a general-purpose business or home computer. The platform became particularly popular as a gaming platform. *blog.yovisto.com

 The comet Hale–Bopp was discovered on July 23, 1995, independently by two observers, Alan Hale and Thomas Bopp, both in the United States. Hale–Bopp's orbital position was calculated as 7.2 astronomical units (AU) from the Sun, placing it between Jupiter and Saturn and by far the greatest distance from Earth at which a comet had been discovered by amateurs. It was discovered at such a great distance from the Sun that it raised expectations that the comet would brighten considerably by the time it passed close to Earth. Although predicting the brightness of comets with any degree of accuracy is very difficult, Hale–Bopp met or exceeded most predictions when it passed perihelion on April 1, 1997. The comet was dubbed the Great Comet of 1997.

The image is by my former student Dan Durda, of Sterling, Michigan; who may have acquired some math knowledge from me, but not his artistic sense.


1773 Sir Thomas Makdougall Brisbane, (23 July 1773 – 27 January 1860) Baronet British soldier and astronomical observer for whom the city of Brisbane, Australia, is named. He was Governor of NSW (1821-25). Mainly remembered as a patron of science, he built an astronomical observatory at Parramatta, Australia, made the first extensive observations of the southern stars since Lacaille in (1751-52) and built a combined observatory and magnetic station at Makerstoun, Roxburghshire, Scotland. He also conducted (largely unsuccessful) experiments in growing Virginian tobacco, Georgian cotton, Brazilian coffee and New Zealand flax.*TIS

1775 Etienne Louis Malus (23 July 1775 – 24 February 1812) born in Paris. He was the son on the Treasurer of France. His primary interest was mathematical optics. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, p. 473] *VFR He studied geometric systems called ray systems, closely connected to Julius Plücker's line geometry. He conducted experiments to verify Christiaan Huygens' theories of light and rewrote the theory in analytical form. His discovery of the polarization of light by reflection was published in 1809 and his theory of double refraction of light in crystals, in 1810.
Malus attempted to identify the relationship between the polarising angle of reflection that he had discovered, and the refractive index of the reflecting material. While he deduced the correct relation for water, he was unable to do so for glasses due to the low quality of materials available to him (most glasses at that time showing a variation in refractive index between the surface and the interior of the glass). It was not until 1815 that Sir David Brewster was able to experiment with higher quality glasses and correctly formulate what is known as Brewster's law.
Malus is probably best remembered for Malus' law, giving the resultant intensity, when a polariser is placed in the path of an incident beam. His name is one of the 72 names inscribed on the Eiffel tower.*Wik

1854 Ivan Vladislavovich Sleszynski (23 July 1854 in Lysianka, Cherkasy, Kiev gubernia, Ukraine - 9 March 1931 in Kraków, Poland)Sleszynski's main work was on continued fractions, least squares and axiomatic proof theory based on mathematical logic. In a paper of 1892, based on his doctoral dissertation, he examined Cauchy's version of the Central Limit Theorem using characteristic function methods, and made several significant improvements and corrections. Because of the work, he is recognised as giving the first rigorous proof of a restricted form of the Central Limit Theorem. *SAU

1856 Bal Gangadhar Tila (23 July 1856 – 1 August 1920, age 64) Scholar, mathematician, philosopher, and militant nationalist who helped lay the foundation for India's independence. Tilak was a great Sanskrit scholar and astronomer. He fixed the origin and date of Rigvedic Aryans, which was highly acclaimed and universally accepted by orientalists of his time. He founded (1914) and served as president of the Indian Home Rule League and, in 1916, concluded the Lucknow Pact with Mohammed Ali Jinnah, which provided for Hindu-Muslim unity in the struggle for independence.*TIS

1886 Walter Schottky (23 July 1886, Zürich, Switzerland – 4 March 1976, Pretzfeld, West Germany) Swiss-born German physicist whose research in solid-state physics led to development of a number of electronic devices. He discovered the Schottky effect, an irregularity in the emission of thermions in a vacuum tube and invented the screen-grid tetrode tube (1915). The Schottky diode is a high speed diode with very little junction capacitance (also known as a "hot-carrier diode" or a "surface-barrier diode.") It uses a metal-semiconductor junction as a Schottky barrier, rather than the semiconductor-semiconductor junction of a conventional diode. *TIS

1906 Vladimir Prelog (23 July 1906 – 7 January 1998) Yugoslavian-born Swiss chemist who shared the 1975 Nobel Prize for Chemistry with John W. Cornforth for his work on the stereochemistry of organic molecules and reactions. Stereochemistry is the study of the three-dimensional arrangements of atoms within molecules. He authored systematic naming rules for molecules and their mirror-image version, that is, which configuration will be referred to as "dextra" and which will be the "levo" (right or left). Also, by X-ray diffraction, he elucidated the structure of several antibiotics.*TIS

1920 Chushiro Hayashi (July 23, 1920 – February 28, 2010) Japanese astrophysicist who with his coworkers created evolutionary models for stars of mass between 0.01 to 100 times that of the Sun. In 1950, he contributed to the abg (Alpher, Bethe, Gamow) (also see April 1, Events) model of nucleosynthesis in the hot big bang. Hayashi pioneered in modeling stellar formation and pre-main sequence evolution along “Hayashi tracks” (1961) downward on the Hertzprung-Russell diagram until stars reach the main sequence. He and Takenori Nakano studied the formation of low-mass, brown dwarf stars. Hayashi also investigated the formation of the solar system and of the earth and its atmosphere. He retired in 1984. He was presented the Bruce Medal in 2004 for lifetime contributions to astronomy.*TIS

1928 Vera Rubin (July 23, 1928 – December 25, 2016) was an American astronomer who pioneered work on galaxy rotation rates. She uncovered the discrepancy between the predicted and observed angular motion of galaxies by studying galactic rotation curves. By identifying the galaxy rotation problem, her work provided evidence for the existence of dark matter. These results were later confirmed over subsequent decades. *Wik

Throughout her schooling, Rubin faced the banal sexism all too common at the time for women interested in science. Her high school physics teacher ignored the few girls in class; a college admissions interviewer encouraged her to consider painting astronomical objects rather than studying them. Rubin was not deterred. “She took it all with courage, and with persistence,” says Bahcall, who was a longtime friend and colleague of Rubin.

Rubin studied astronomy at Vassar College, married mathematical physicist Bob Rubin, and began graduate school at Cornell in 1948. She completed her master’s thesis on the rotation of the universe, kicking off a long career investigating hidden galactic behavior, and a professor suggested Rubin present this research at a meeting of the American Astronomical Society (AAS) in 1950. There, she encountered a chilly reception. “I gave my memorized 10-minute talk, acceptably I thought. Then one by one many angry sounding men got up to tell me why I could not do ‘that,’” she wrote.
In 2019, almost 70 years after Rubin faced that hostile crowd at the 1950 AAS meeting, (and almost three years after her Death) the Society convened in Honolulu to agree on renaming the Large Synoptic Survey Telescope as the Vera C. Rubin Observatory. After its construction is complete, the Rubin Observatory will be the site of a 10-year survey of a mammoth swath of night sky. Bahcall sees this honor as the perfect tribute to Rubin, who cherished the many long nights she spent in observatories, peering through telescopes. “She loved being in the dome.” *APS Org

1930 Daniel McCracken, (July 23, 1930 – July 30, 2011) who wrote the first textbook on FORTRAN, was born. A student of mathematics and chemistry, McCracken started working in computers at General Electric in 1951, training workers in using the new technology. Based on this teaching experience, McCracken wrote several important computer programming textbooks, most notably ""A Guide to FORTRAN Programming"" in 1961.*CHM

1932 Derek Thomas "Tom" Whiteside FBA (July 23, 1932–April 22, 2008[4]) was a British historian of mathematics. He was the foremost authority on the work of Isaac Newton and editor of The Mathematical Papers of Isaac Newton. From 1987 to his retirement in 1999, he was the Professor of the History of Mathematics and Exact Sciences at Cambridge University. *Wik

1941  Pierre Agostini (born 23 July 1941, ) is a French experimental physicist and Emeritus professor at the Ohio State University in the United States, known for his pioneering work in strong-field laser physics and attosecond science.He is especially known for the observation of above-threshold ionization and the invention of the reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) technique for characterization of attosecond light pulses. He was jointly awarded the 2023 Nobel Prize in Physics with Ferenc Krausz and Anne L’Huillier. 

1952 Mark David Weiser (July 23, 1952 – April 27, 1999) American computer scientist and visionary who developed the pioneering idea for what he referred to as "ubiquitous computing," He coined that term in 1988 to describe a future in which PC's will be replaced with tiny computers embedded in everyday "smart" devices (everyday items such as coffeepots and copy machines) and their connection via a network. He said, "First were mainframes, each shared by lots of people. Now we are in the personal computing era, person and machine staring uneasily at each other across the desktop. Next comes ubiquitous computing, or the age of calm technology, when technology recedes into the background of our lives." *TIS


1876  Joseph Rogers Brown (born Jan. 26, 1810, Warren, R.I., U.S.—died July 23, 1876, Isles of Shoals, N.H.) was an American inventor and manufacturer who made numerous advances in the field of fine measurement and machine-tool production. He perfected and produced a highly accurate linear dividing engine in 1850, and in the succeeding two years he developed a vernier caliper reading to thousandths of an inch and also applied vernier methods to the protractor. Brown's micrometer caliper, widely used in industry, appeared in 1867. He also invented a precision gear cutter in 1855 to produce clock gears, a universal milling machine in 1862, and, perhaps his finest innovation, a universal grinding machine (patented in 1877), in which articles were hardened first and then ground, thereby increasing accuracy and eliminating waste. Cofounded J.R. Brown & Sharpe in 1853. *TiS

During the 19th and 20th centuries, Brown & Sharpe was one of the best-known and most influential machine tool builders and was a leading manufacturer of instruments for machinists (such as micrometers and indicators). Its reputation and influence were such that its name is often considered to be inseparably paired with certain industrial standards that it helped establish, including:

The American wire gauge (AWG) standards for wire;

The Brown & Sharpe taper in machine tool spindle tapers; and

The Brown & Sharpe worm threadform for worm gears.

Many years ago as a young Gauge Lab Tech I had a drawer full of these beauties in various sizes.  

1903 Eduard Weyr (22 June 1852 Praha – 23 July 1903 Záboří) wrote geometrical papers and books mainly in projective geometry and differential geometry. He also worked on algebra, in particular studying linear algebra, matrices and hypercomplex systems.
Weyr published Differential calculus in 1902. This led to controversy with a young mathematician J V Pexider who sharply criticised Weyr's textbook. Jindrich Beèváo and Ludek Zajièek give an interesting account of this episode in a paper in the book .*Math.info website

1916 William Ramsay (2 October 1852, Glasgow, Scotland - 23 July 1916 (aged 63)
High Wycombe, Bucks., England) died. Ramsay was a British chemist who discovered the four gases neon, argon, krypton and xenon. He also determined they belonged with helium and radon to form a family of gases called the noble gases. This discovery would earn him the 1904 Nobel Prize in Chemistry.*Science History

1932 Alberto Santos-Dumont (July 20, 1873 – July 23, 1932) was a Brazilian aviation pioneer, deemed the Father of Aviation by his countrymen. At the age of 18, Santos-Dumont was sent by his father to Paris where he devoted his time to the study of chemistry, physics, astronomy and mechanics. His first spherical balloon made its first ascension in Paris on 4 July 1898. He developed steering capabilities, and in his sixth dirigible on 19 Oct 1901 won the "Deutsch Prize," awarded to the balloonist who circumnavigated the Eiffel Tower. He turned to heavier-than-air flight, and on 12 Nov 1906 his 14-BIS airplane flew a distance of 220 meters, height of 6 m. and speed of 37 km/h. to win the "Archdecon Prize." In 1909, he produced his famous "Demoiselle" or "Grasshopper" monoplanes, the forerunners of the modern light plane. *TIS

Santos-Dumont circling the Eiffel Tower with the airship No. 5, 13 July 1901

1964 Samarendra Nath Roy or S. N. Roy (11 December 1906 – 23 July 1964). He was well known for his pioneering contribution to multivariate statistical analysis, mainly that of the Jacobians of complicated transformations for various exact distributions, rectangular coordinates and the Bartlett decomposition. His dissertation included the Post master's work at the Indian Statistical Institute where he worked under Mahalanobis. To commemorate his Birth Centenary an International Conference on "Multivariate Statistical Methods in the 21st Century: The Legacy of Prof. S.N. Roy" was held at Kolkata, India during December 28–29, 2006. The Journal of Statistical Planning and Inference published a special Issue for celebrating of the Centennial of Birth of S. N. Roy*Wik

1964 W. W. Rogosinski died. *VFR He wrote on Fourier Series with G.H. Hardy

1993 Florence Nightingale David, (August 23, 1909 – July 23, 1993) also known as F. N. David was an English statistician, born in Ivington, Herefordshire, England. She was named after Florence Nightingale, who was a friend of her parents.
David read mathematics at Bedford College for Women in London. After graduation, she worked for the eminent statistician Karl Pearson at University College, London as his research student. She calculated the distribution of correlation coefficients, producing in 1938 her first book, Tables of the correlation coefficient.
After Karl Pearson died in 1934, she returned to the Biometrics laboratory to work with Jerzy Neyman where she submitted her last four published papers as her PhD thesis. During World War II, David worked for the Ministry of Home Security. In late 1939 when war had started but England had not yet been attacked, she created statistical models to predict the possible consequences of bombs exploding in high density populations such as the big cities of England and especially London. From these models, she determined estimates of harm to humans and damage to non-humans This included the possible numbers living and dead, the reactions to fires and damaged buildings as well as damages to communications,utilities such as phones, water, gas, electricity and sewers. As a result when the Germans bombed London in 1940 and 1941, vital services were kept going and her models were updated and modified with the evidence from the real harms and real damage.
David became head of the Statistics Department at the University of California at Riverside in 1970.*Wik

2012  Sally Kristen Ride (May 26, 1951 – July 23, 2012) was an American physicist and a former NASA astronaut. Ride joined NASA in 1978, and in 1983 became the first American woman—and then-youngest American, at 32—to enter space. In addition to being interested in science, she was a nationally ranked tennis player. Ride attended Swarthmore College and then transferred to Stanford University, graduating with a bachelor's degree in English and physics. Also at Stanford, she earned a master's degree and a Ph.D. in physics, while doing research in astrophysics and free electron laser physics.In 1987 she left NASA to work at Stanford University's Center for International Security and Arms Control.
Sally Ride died on July 23, 2012 after a 17-month battle with pancreatic cancer.   US President Barack Obama called her a "national hero and a powerful role model" who "inspired generations of young girls to reach for the stars."*Wik

1933 Steven Weinberg (May 3, 1933 – July 23, 2021) American nuclear physicist who shared the 1979 Nobel Prize for Physics (with Sheldon Lee Glashow and Abdus Salam) for work in formulating the electroweak theory, which explains the unity of electromagnetism with the weak nuclear force. *TIS

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Monday 22 July 2024

Kurschak's Theorem, a Geometry Jewel.

József Kürschák (14 March 1864 – 26 March 1933) was a Hungarian mathematician noted for his work on trigonometry, but he may be more well known today, by the very few who seem to know of him, for a geometric gem that involves no Trig.  The basic idea is to prove , without trig, that the area of a regular dodecagon inscribed in a circle with unit radius is exactly three.  [a footnote about unusual things to share with students; if you inscribe a regular polygon in a unit circle, there are only two of them that will have a rational area.  I just gave away one of them, the other should not be too difficult to discern.]

The classical approach for any regular polygon is the one used by MathWorld.  

The area of the dodecagon (n=12) inscribed in a unit circle with R=1 is

 \(A=\frac{n}{2} R^2 sin(\frac{2\pi}{n})=3. \)

[Teachers may discuss among themselves the appropriate deduction for a correct result that insults the entire idea of the theorem...?]

So how did he do it?  Well the essence of the pretty image at the top may mask some of the deep ideas so let's part the haze a little.  This diagram shows all the elements of the solution.  

Equlateral triangles are constructed on the sides of a square inwardly. Their apexes form a square. Prove that the midpoints of the sides of the latter and the intersections of the side lines of the triangles form a dodecagon.

The inner square is the Kurschak Tile, and the source of a second name for the image.You can observe in the image that dodecagon is tiled by 24 isosceles 15°-15°-150° triangles and 12 regular triangles. Two isosceles triangles sharing the base combine into a rhombus with the side equal to that of regular triangles.

The area of the square outside the dodecagon is composed of 8 isosceles triangles identical to the ones forming the Rhombl in the interior of the dodecagon, and an additional 8 regular triangles congruent to the ones inside.  

And now we apply two easy images fromKürschak's Tile, G. L. Alexanderson, Kenneth Seydel; The Mathematical Gazette, Vol. 62, No. 421 (Oct., 1978), pp. 192-196 

We first realize that if the circle has a radius of one, thevequilateral triangles that formed the original construction, and the tile they formed, have edges of 2 units, Thus the whole tile has an area of four, made up of four unit squares.

In one of those unit squares there are nine triangles that are inside that square so we mark them to remove.  


Now we look at the parts of the dodecagon in the other three squares and note that there are some blank spaces outside the dodecagon.  And in an instant you recognize that all nine of the squares from the interior of the dodecagon will nicely fill each of the missing parts outside the dodecagon in these three squares, filling a space of three square units.  

And as the cool kids say, QED

Added notes and images from *Cut -the-knot.org , *  Wolfram MathWorld.  and 

"It could easily be shown..." Probability and Pi and the Riemann Zeta Function

 Rushed re-edit , Fingers crossed

Beware of articles that begin, "It could easily be shown..." It is like arm wrestling a two year old, if you win, so what, and if you LOSE??? Yow....
I know this, and yet, I still proceed foolishly to read them. The one currently on my mind was a "Note on Pi" by R. Chartes in the March 1904 Philosophical Magazine, my current old document of choice. It pointed out that ICEBS "that if two numbers are written down at random, the probability that they will be prime to each other is 6/pi2."
Here it is from Wolfram Mathworld:

This is the reciprocal of the famous answer to the Basel problem evaluate by Euler.

The fact that the probability that two random numbers are relatively prime was equal to this value was discovered by M. Cesaro and J. J. Sylvester in the same year, 1883. Sylvester gives a proof in a footnote to a paper I found in his collected works (page 602)

The original image from Sylvester seems to have dissappeared into the abyss since 2009, so I share a proof I found on the Physics Harvard edu websight,

Ok, yeah, that is sort of easy, and I should have figured it out...The proof is easy to extend to the probability that three numbers are relatively prime is the reciprocal of the sum of the reciprocal of the cubes (if that seems hard to read, try to write it). More simply, the probability is the reciprocal of ζ(3)=

 Let  a,b and c be integers chosen at random

The probability that a, b, and c have no common divisor:


where 𝜁 denotes the zeta function:

the decimal value is approximately 

 Strangely, the discovery (by Sylvester) is nested in work he was doing with Farey Fractions.  

If you haven't been exposed to Farey Fractions, a quick share from Wolfram's Mathworld

The Farey sequence F_n for any positive integer n is the set of irreducible rational numbers a/b with 0<=a<=b<=n and (a,b)=1 arranged in increasing order. The first few are


(OEIS A006842 and A006843). Except for F_1, each F_n has an odd number of terms and the middle term is always 1/2.

The image at the top shows a pair of coordinate axes with a point (x,y) painter black if GCF(m,n)=1, and white otherwise.

For students, it should be made clear that these methods apply to random numbers up to a very large n. They should understand that these are the probabilities as the range of the random selections approach infinity.  
They should also realize that if one of the numbers is prime, the pair will always be co-prime.  As the numbers get very large, the percentage of the primes gets smaller.  A good general rule for the number of primes that are equal to or less than n approaches x/ ln (n) as n approaches infinity. For smaller numbers they should be helped to realize that this limit usually under-counts the number of primes.  For n=100, for instance, 100/ln(100) =21.7... but there are actually 25 prime numbers less than 100. As n gets larger the ratio of primes to predicted primes grows much closer to 1.  (A great use of spreadsheets to let them explore on their own.  The top graph below shows this error.

I think one of the really nice things that can be done with younger students studying common factors and slope (can I say in Alg I?) is to show them that the greatest common factor of m and n is the number of lattice points on the line from (0,0) to (m,n)....[not counting (0,0)] Here is a graph of the segments to (4,10) and (12,3)