Tuesday 18 June 2024

On This Day in Math - June 18


I began to understand that pure mathematics was more than a collection of random tools mainly fashioned for use in the Cambridge treatment of natural philosophy.

Andrew Forsyth

The 169th day of the year; 169 is the smallest square which is prime when rotated 180o (691)  What is the next one?

And from Jim Wilder, 169 is the reverse of 961. The same is true of their square roots... √169=13 and √961=31
or stated another way, 169 = 132 and in reverse order 312 = 961

An interesting loop sequence within Pi. If you search for 169, it appears at position 40. If you then search for 40, it appears at position 70. Search for 70, ... 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, *Pi Search page

169 is the only year day which is both the difference of consecutive cubes, and a square: \(8^3-7^3 =169=13^2\)

The first successful dissection of a square into smaller squares was of a square with 169 units on a side. 1907-1914 S. Loyd published The Patch Quilt Puzzle. A square quilt made of 169 square patches of the same size is to be divided into the smallest number of square pieces by cutting along lattice lines. The answer, which is unique, is composed of 11 squares with sides 1,1,2,2,2,3,3,4,6,6,7 within a square of 13. It is neither perfect nor simple. Gardner states that this problem first appeared in 1907 in a puzzle magazine edited by Sam Loyd. David Singmaster lists it as first appearing in 1914 in Cyclopedia by Loyd but credits Loyd with publishing Our Puzzle Magazine in 1907 - 08. This puzzle also appeared in a publication by Henry Dudeney as Mrs Perkins Quilt. Problem 173 in Amusements in Mathematics. 1917


In 1178, about an hour after sunset - as chronicled by the English monk, Gervase of Canterbury - a band of five eyewitnesses watched as the upper horn of the bright, new crescent Moon "suddenly split in two. From the midpoint of this division a flaming torch sprang up, spewing out... fire, hot coals and sparks... The body of the moon, which was below writhed... throbbed like a wounded snake." In 1976, a geologist suggested that this was consistent with the location and age of the 22-km lunar crater Giordano Bruno. However, such asteroid impact would have ejected debris causing an astonishing meteor shower, which was never reported. Now the sighting of 1178 is attributed to perhaps an exploding meteor that just happened to line up with their view of the Moon. *TiS

1558 Robert Recorde’s will was admitted to probate, after he died in prison. He introduced the equals sign in The Whetstone of Witte (1557) with the words: “And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lenghte, thus:  because noe .2. thynges, can be moare equalle.” “Gemowe” (think Gemini )is an old French work meaning “twin.”. *VFR  When they are asked what they would use if this was not available, it seems difficult for students to imagine a different symbol.  

Image from Wikipedia.

1584 Jacob Christmann appointed professor of Hebrew at Heidelberg. In 1595 he defended the view that the circle could only be approximately squared. *VFR

A De revolutionibus manuscript of Nicolaus Copernicus passed via Rheticus to others and was marked on 19 December 1603 by Christmann with Nicolai Copernick Canonici Varmiensis in Borussia Germaniae mathematici … ("of Canon Nicolaus Copernick from Warmia in Prussia of Germany, of the mathematician …"). Since 1953 it is located in Kraków in the Jagiellonian library (Signatur: Ms. BJ. 10 000) and is accessible online. *Wik

Note by Christmann in the De revolutionibus manuscript, 1603

1791 Denison Olmsted (June 18, 1791 – May 13, 1859) was an American physicist and astronomer. Professor Olmsted is credited with giving birth to meteor science after the 1833 Leonid meteor shower over North America spurred him to study this phenomenon.
Olmsted and his associate, Elias Loomis, were in 1835 the first American investigators to observe the Halley's Comet.
Olmsted possessed considerable mechanical talent, which he used in promoting and perfecting the inventions of others, but while he himself frequently invented articles of convenience and comfort, such as the Olmsted stove, he seldom secured his rights by patents. *Wik

Olmsted is best known for his studies of the Great Meteor Storm of 1833. The event occurred over much of North America, and at its peak, some 100,000 meteors per hour were streaking down toward earth.  Olmsted observed the storm and noted that the meteors radiated from a point in the constellation Leo.  He further noted that the radiating point moved with the stars, suggesting that the meteors had a cosmic origin.  Olmsted sent out a call for more observations to his colleagues in America and Europe, and he published those various reports in the American Journal of Science (conveniently published at Yale) in late 1833.  In a follow-up article in the next issue of the Journal, he published his conclusions about the extra-terrestrial origins of meteor storms.  This marks the birth of the study of meteor showers.  The storm of 1833 was an occurrence of the annual Leonid meteor shower, named for the radiating point from which the meteors stream, and that Olmsted determined.  The Leonids appear every November, but they are especially profuse every 33 years.  1833 was one of those bonus years. We have the two journal articles that Olmsted wrote in our serials collection.  I read them out of curiosity, and I was struck with how much garbage Olmsted had to wade through in trying to understand the nature of the meteor shower.  Some observers noted that the temperature dropped drastically after the storm; that high winds arose; that auroras increased; and many claimed that the radiating point moved with the earth, not the stars.  Somehow Olmsted managed to sift through all this, discard the dross, and come up with the essential facts. 
It is not so well known that Olmsted was a gifted writer of popular astronomy texts.  He published Letters on Astronomy, Addressed to a Lady in 1840; the book was then enlarged and reprinted as The Mechanism of the Heavens (1850).  We do not have the first work, but we do have the 1850 and 1853 editions of The Mechanism of the Heavens.*Linda Hall Org

1846 William Lassell, an English amateur astronomer, was one of the first to use what we call an equatorial mount for his telescopes, where the main axis of the mount parallels the earth's axis. This means that if you rotate the telescope slowly around this axis, you can in effect make the earth stand still, and likewise the stars. Lassell first built a 24", 20-foot reflector (meaning the mirror was two feet across and the tube was 20 feet long).  We have no image of that telescope in its original location, but much later it was given to the Royal Observatory at Greenwich, where they mounted it in a dome, called the Lassell Dome, and we see a drawing of it made around 1890.
With his 20-footer, Lassell discovered Triton, the large moon of Neptune, on Oct. 10, 1846. This was just 17 days after Neptune itself was discovered, and Triton is less than one-tenth the diameter of Neptune, which indicates just how good Lassell's telescope was. Using the same telescope, Lassell in 1851 discovered two new moons around Uranus, to go with the two that William Herschel had found shortly after he discovered Uranus in the first place.  William’s son John Herschel named Lassell’s new moons, Ariel and Umbriel, after two sprites in Shakespeare’s The Tempest and Alexander Pope’s The Rape of the Lock. 

*Linda Hall Org

1864 Lewis Carroll finally decided to write up Alice’s Adventures in Wonderland. [Stuart Dodgson Collingwook, The Life and Letters of Lewis Carroll (1898), p. 96]
In 1862 he went boating on the Isis, a tributary of the Thames, with the three daughters of Henry George Liddell, dean of Christ Church, Oxford. He was especially fond of Alice Liddell, then ten, and it was mainly for her that he began the story of another Alice’s tumble down a rabbit hole.
Image: The Original 1865 Edition

1908  Alan Archibald Campbell Swinton took the first x-ray images in Britain in January 1896 and by a year later the medical professions were bringing him surgical cases for analysis. But "on this day he predicted exactly how another magic box would work, in a letter to Nature. He called it ‘Distant Electric Vision’, but we know it now as television." *Keith Moore, http://blogs.royalsociety.org

Responding to an article in the June 4, 1908 issue of Nature by Shelford Bidwell entitled "Telegraphic Photography and Electric Vision," [also included here in original wrappers] A. A. Campbell Swinton wrote a letter to the editor of Nature proposing a solution to the most pressing problems in achieving "distant electric vision":

"The final, insurmountable problems with any form of mechanical scanning were the limited number of scans per second, which produced a flickering image, and the relatively large size of each hole in the disk, which resulted in poor resolution. In 1908 a Scottish electrical engineer, A. A. Campbell Swinton, wrote that the problems 'can probably be solved by the employment of two beams of kathode rays' instead of spinning disks. Cathode rays are beams of electrons generated in a vacuum tube. Steered by magnetic fields or electric fields, Swinton argued, they could 'paint' a fleeting picture on the glass screen of a tube coated on the inside with a phosphorescent material. Because the rays move at nearly the speed of light, they would avoid the flicker problem, and their tiny size would allow excellent resolution. Swinton never built a set (for, as he said, the possible financial reward would not be enough to make it worthwhile)..." (Britannica).

Swinton presciently concluded his letter with the hopeful statement that with his technique "distant electric vision will, I think, come within the region of possibility." Cathode rays would, of course, be used to great success in the development of television and Swinton's recognition of their possibilities is often cited as the critical discovery that made television possible. *Manhattan Rare Books 

 1928, aviator Amelia Earhart became the first woman to fly across the Atlantic Ocean. She had accepted the invitation of the American pilots Wilmer Stultz (1900-29) and Louis Gordon to join them on the transatlantic flight. The crossing from Newfoundland to Wales took about 21 hours. Amelia Earhart went on to establish herself as a respected role model, tirelessly demonstrating that young women were as capable as men in succeeding in their chosen vocations. In 1935 she crossed the Atlantic solo in record time: 13 hr 30 min.  *TIS

1983 Sally Ride, astrophysicist, becomes the first American woman in space. The Soviets were ahead by twenty years and two days.*VFR
She was selected as a mission specialist astronaut with NASA Astronaut Group 8, the first class of NASA astronauts to include women. After completing her training in 1979, she served as the ground-based capsule communicator (CapCom) for the second and third Space Shuttle flights, and helped develop the Space Shuttle's robotic arm. In June 1983, she flew in space on the Space Shuttle Challenger on the STS-7 mission. The mission deployed two communications satellites and the first Shuttle pallet satellite (SPAS-1). Ride operated the robotic arm to deploy and retrieve SPAS-1. Her second space flight was the STS-41-G mission in 1984, also on board Challenger. She spent a total of more than 343 hours in space. She left NASA in 1987. *Wik


1799 William Lassell (18 June 1799 – 5 October 1880) was a wealthy amateur English astronomer. He set up an observatory at Starfield, near Liverpool. England, He built his own 24" diameter telescope, and devised steam-driven equipment for grinding an polishing the speculum metal mirror. This telescope was the first of its size to be mounted "equitorially" to allow easy tracking of the stars. He discovered Triton, a moon of Neptune, and Ariel and Umbriel, satellites of Uranus. Later, Lassell built a 48" diameter telescope with the same design and took it to Malta for observations with clearer skies.*TIS  

Lassel’s 37-foot reflector was the last of the great "speculum" reflectors, where the mirror was ground from a metal alloy called speculum, made mostly of copper and tin. All future large mirrors would be ground from glass, with a silver reflecting surface added as a thin film. Silvered-glass mirrors would be cheaper, lighter, brighter, and much less susceptible to tarnishing. Lassell was also the last of the great gentleman amateur astronomers, as astronomy became ever more professionalized in the late 19th century. *Linda Hall org

1818 Pietro Angelo Secchi (18 Jun 1818, 26 Feb 1878 at age 59) Italian Jesuit priest and astrophysicist, who made the first survey of the spectra of over 4000 stars and suggested that stars be classified according to their spectral type. He studied the planets, especially Jupiter, which he discovered was composed of gasses. Secchi studied the dark lines which join the two hemispheres of Mars; he called them canals as if they where the works of living beings. (These studies were later continued by Schiaparelli.) Beyond astronomy, his interests ranged from archaeology to geodesy, from geophysics to meteorology. He also invented a meteorograph, an automated device for recording barometric pressure, temperature, wind direction and velocity, and rainfall.*TIS

1858 Andrew Russell Forsyth (18 June 1858, Glasgow – 2 June 1942, South Kensington) studied at Liverpool College and was tutored by Richard Pendlebury before entering Trinity College, Cambridge, graduating senior wrangler in 1881. He was elected a fellow of Trinity and then appointed to the chair of mathematics at the University of Liverpool at the age of 24. He returned to Cambridge as a lecturer in 1884 and became Sadleirian Professor of Pure Mathematics in 1895. *Wik 
In 1893 he published Theory of functions of a complex variable which had such an impact at Cambridge that function theory dominated there for many years. Whittaker writes... that this text:-
... had a greater influence on British mathematics than any work since Newton's Principia.
However the reputation of the book outside Britain was not high. In fact this is not surprising since the whole thrust of the book was to bring the great advances of Continental mathematics to Cambridge which Forsyth rightly saw as living in the past. He was well equipped to undertake this task for he traveled widely and, being a good linguist, was able to appreciate the advances made by authors writing in French and German.
On Cayley's death Forsyth was appointed to his chair in 1895 becoming the Sadleirian professor of Pure Mathematics. However his preference for technical mastery rather than rigorous analysis meant that he failed to inspire future pure mathematicians. In fact one would have to say that Forsyth was unlucky, for although he saw the importance of Continental mathematics, at the same time his greatest strengths lay in his ability to handle complex formulae. He therefore excelled at precisely the style of mathematics which he himself campaigned successfully to replace at Cambridge.

1884  Charles Weatherburn  (18 June, 1884 in Australia - 18 October,1974 in Australia) worked on vector analysis and differential geometry.*SAU
Weatherburn graduated from the University of Sydney an MA in 1906. After being awarded a scholarship he studied at Trinity College, Cambridge sitting the Mathematical Tripos examinations in 1908. Weatherburn was awarded a First Class degree. On his return to Australia, Weatherburn taught at Ormond College of the University of Melbourne.

In 1923 was appointed chair of mathematics in Canterbury College, University of New Zealand. He returned to Australia in 1929 as chair of mathematics at the University of Western Australia, a post he held until he retired in 1950.

He died in Perth, Western Australia in 1974.

1884 Frieda Nugel (18 June 1884 in Cottbus, Brandenburg, Germany- 6 Nov 1966 in Bad Godesberg, Bonn, Germany) was a German mathematician, and civil rights activist, who was one of the first women to receive a doctorate in Germany *SAU  
She earned her PhD at Martin Luther University of Halle-Wittenberg in 1912, under the supervision of August Gutzmer.
She took a position as a teacher at a girls' school in Cottbus, but taught there only for two years, until her marriage to Louis Hahn in 1914. 
In 1927, after the collapse of her husband's newspaper business, Nugel obtained a part-time position at a school in Emden. By 1930 her position there had become permanent, albeit at a smaller salary than the men in her school. The subjects she taught during this time included mathematics, physics, and German. Between 1939 and 1945 she witnessed the bombing of the city of Emden, as part of WWII, and the school was forced to move to Bad Wildungen; her two sons served as officers in the war, and were both killed in 1944.
Nugel retired in 1945 at the age of 61. Her husband died of an illness in 1952. In 1955, she moved to Bad Godesberg in order to connect with her remaining family. In 1962, the Faculty of Mathematics and Natural Sciences at Halle gave her a "Golden Doctoral Diploma" award, on the 50th anniversary of her 1912 dissertation. She died on 6 November 1966 in the town of Bad Godesberg.*Wik

1913  Oswald Teichmüller's (June 18, 1913 – September 11, 1943)  main contribution is in the area of geometric function theory.
To catalog under the "Smart guys are not always good guys category, 

On 2 November 1933 Teichmüller led the student boycott of Edmund Landau's lectures. As background to this we note that Hitler had come to power on 30 January 1933 and on 7 April of that year the 'Law for the reorganisation of the Civil Service' was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Before any official word reached Göttingen from the Ministry, the Dean wrote to Landau on 28 April asking him not to give his summer lecture courses and these were given instead by Landau's assistant Werner Weber. Having received no further advice from the university authorities, Landau decided to give his autumn lectures as advertised. Landau described in unemotional terms what happened on the first day of lectures.

"On 2 November, about 11.15, as I wished to leave my office and go to the large lecture theatre to begin my lecture, the entrance hall was filled with about 80 to 100 students who let me pass through unhindered. In the lecture hall was one person. Clearly therefore, there was a boycott with sentries at the door who had prevented (without force) those students who wanted to work from setting foot in the lecture room."
Teichmüller, as the leader of the boycott, went to Landau's office and discussed what had happened. Landau requested that Teichmüller put his views in writing and he did so. A translation of part of this letter is given

"Through yesterday's action a completely new situation has now been created. In order to restore peace in our institute it is necessary, above all, to clear up the fundamentals behind it. You spoke of your belief that what happened yesterday was an anti-Semitic demonstration. My standpoint was, and continues to be, that an anti-Jewish individual action might rather be directed against everyone else than against you. I am not concerned with making difficulties for you as a Jew, but only with protecting - above all - German students of the second semester from being taught differential and integral calculus by a teacher of a race quite foreign to them. I, like everyone else, do not doubt your ability to instruct suitable students of whatever origin in the purely abstract aspects of mathematics. But I know that many academic courses, especially the differential and integral calculus, have at the same time educative value, inducting the pupil not only to a conceptual world but also to a different frame of mind. But since the latter depends very substantially on the racial composition of the individual, it follows that an Aryan student should not be allowed to be trained by a Jewish teacher."

Chowdhury comments :-
"I find it an extraordinary piece of writing, shamelessly upholding an indefensible attitude and an ignominious action, wherein the brilliant but thoroughly indoctrinated mind of the writer shines through.
We note that Teichmüller had some success in converting others to the Nazi beliefs. For example he convinced Landau's assistant Werner Weber so that he joined the Nazi Party in 1933."

1915 Alice Turner Schafer (June 18, 1915 – September 27, 2009) was an American mathematician. She was one of the founding members of the Association for Women in Mathematics in 1971.

For three years Alice was a secondary school teacher, accruing savings to pay for graduate school.

At University of Chicago, Alice was a student of Ernest Preston Lane, author of Metric Differential Geometry of Curves and Surfaces (1940) and A Treatise on Projective Differential Geometry (1942). Alice studied differential geometry of curves and implications of the singular point of a curve. When a curve has null binormal, it is planar at that point. Duke Mathematical Journal published her work in 1944. Alice continued her investigations into curves near an undulation point, publishing in American Journal of Mathematics in 1948.

When she was completing her studies at Chicago, she met Richard Schafer, who was also completing his Ph.D. in mathematics at Chicago. In 1942 Turner married Richard Schafer, after both had completed their doctorates. They had two sons.

After completing her Ph.D., Alice Schafer taught at Connecticut College, Swarthmore College, the University of Michigan and several other institutions. In 1962 she joined the faculty of Wellesley College as a full professor. Her husband Richard was working at the Massachusetts Institute of Technology, researching non-associative algebras. In 1966 he published a book on them which he dedicated "To Alice".

As a teacher, Alice especially reached out to students who had difficulties with or were afraid of mathematics, by designing special classes for them. She took a special interest in helping high-school students, women in particular, achieve in mathematics.

In 1971, Schafer was one of the founding members of the Association for Women in Mathematics. She was elected as the second President of the Association. "Under the leadership of its second president Alice T. Schafer, [AWM] was legally incorporated in 1973 and received tax-exempt status in 1974.

Schafer was named Helen Day Gould Professor of Mathematics at Wellesley in 1980. She retired from Wellesley in 1980. However, she remained there for two more years during which she was chairman of Wellesley's Affirmative Action Program. After retiring from Wellesley, she taught at Simmons College and was also involved in the management program in the Radcliffe College Seminars. Her husband retired from MIT in 1988 and the couple moved to Arlington, Virginia. However, she still wanted to teach. She became professor of mathematics at Marymount University until a second retirement in 1996.

In 1990 the Association for Women in Mathematics established the Alice T. Schafer Prize in Mathematics to honor her for her dedicated service towards increasing the participation of women in mathematics.

1926 Allan Rex Sandage  (June 18, 1926 – November 13, 2010)  U.S. astronomer who (with Thomas A. Matthews) discovered, in 1960, the first optical identification of a quasi-stellar radio source (quasar), a starlike object that is a strong emitter of radio waves. Although a strange source of radio emission, in visible light, it looked like a faint star. Yet this object was emitting more intense radio waves and ultraviolet radiation than a typical star. He is best known for determining the first reasonably accurate value for the Hubble constant and the age of the universe.*TIS & Wik


1818 George Baron (?? , June 18, 1818) was a mathematician who emigrated from Northumberland, England to Hallowell, Maine in the United States, thereafter moving to New York. He was the first superintendent and mathematics professor at what would become the United States Military Academy in 1801 and the founder and editor-in-chief of the Mathematical Correspondent, which was the first American "specialized scientific journal" and the first American mathematics journal, first published May 1, 1804.  The journal published an essay by Robert Adrian which was the first to introduce Diophantine analysis in the United States. In 1807, Adrian, a main contributor to the journal, became editor for one year.  Adrain also published a paper on the normal law of errors in 1808, one year before Gauss.

Baron was first offered the position at the fledgling academy at West Point, New York by the newly elected United States President Thomas Jefferson's Secretary of War Henry Dearborn, a friend of Baron's who had lived near him in Maine. After agreeing upon salary and perks, instruction began on September 21, 1801 employing the use of Charles Hutton's A Course in Mathematics and a blackboard, the first recorded use of the latter in America. In October, there was a disagreement between Baron and one of the cadets, Joseph Gardner Swift. Swift was called upon to apologize and was reprimanded for the language he employed against Baron, but went on to become the Military Academy's first graduate, and later a Brigadier General. For a variety of reasons, Baron was court-martialled in December, and Major Jonathan Williams became the supervisor and Captain William Amherst Barron became the instructor of mathematics.
Baron became a teacher of mathematics in New York City, there joining the Theistical Society of New York, a deist group led by Elihu Palmer that came to public attention in the course of a pamphlet war between supporters of United States Vice President Aaron Burr and supporters of then United States Senator from New York DeWitt Clinton. *Wik
Baron may have been one of the earliest users of a blackboard in the US as, "use of the blackboard was a favorite method of Baron." *Edward S Holden, The Centenial of the US Military Academy at West Point, New York

1922 Jacobus Cornelius Kapteyn(January 19, 1851, Barneveld, Gelderland – June 18, 1922) Dutch astronomer who used photography and statistical methods in determining the motions and spatial distribution of stars. Such work was the first major step after the works of William and John Herschel. He tried to solve the questions of space density of stars as a function of distance from the sun, and the distribution of starts according to brightness per unit volume. Some of his results had lasting value, but some were superceded because he had failed to account for the interstellar absorption. In studies using proper motion to determine stellar distances, he discovered stellar motions are not random, as previously thought, but that stars move in two "star streams" (1904). He introduced absolute magnitude and colour index as standard concepts.*TIS

1935 Alexander von Brill (20 September 1842 – 18 June 1935) died. He worked on algebraic geometry and the theory of algebraic functions.  Born in Darmstadt, Hesse, he attended University of Giessen where he earned his doctorate under supervision of Alfred Clebsch. He held a chair at the University of Tübingen, where Max Planck was among his students.
The London Science Museum contains sliceform objects prepared by Brill and Felix Klein.

1862 Florence Bascom (July 14, 1862 – June 18, 1945) was an American pioneer for women as a geologist and educator. Bascom became an anomaly in the 19th century when she earned two bachelor's degrees. Earning a Bachelor of Arts in 1882, and a Bachelor of Science in 1884 both at the University of Wisconsin. Shortly after, in 1887, Bascom earned her master's degree in geology at the University of Wisconsin. Bascom was the second woman to earn her PhD in geology in the United States, in 1893. Receiving her PhD from Johns Hopkins University, this made her the first woman to earn a degree at the institution. After earning her doctorate in geology, in 1896 Bascom became the first woman to work for the United States Geological Survey as well as being one of the first women to earn a master's degree in geology. Bascom was known for her innovative findings in this field, and led the next generation of female geologists. Geologists consider Bascom to be the "first woman geologist in America". 

By 1924, Bascom became a councillor of the Geological Society of America and in 1930 she was appointed as vice-president of that society making her the only woman to have ever held those offices. Bascom's career consisted of her being an editor of the American Geologist, a member of the National Academy of Sciences, the National Research Council, as well as the Geophysical Union and many other scientific societies.

1980 Kazimierz Kuratowski  (February 2, 1896 – June 18, 1980) He worked in the area of topology and set theory. He is best known for his theorem giving a necessary and sufficient condition for a graph to be planar.*SAU   
Kuratowski's theorem:  "A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or K3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three)."  (in simpler, but less exact terms,  it can be drawn in such a way that no edges cross each other."  The well-known recreational problem of connecting three houses to three utilities is not possible to draw because it is K3,3 (below).  The utility problem posits three houses and three utility companies--say, gas, electric, and water--and asks if each utility can be connected to each house without having any of the gas/water/electric lines/pipes pass over any other. (1913 Dudeney: first publication of Gas, Water and Electricity Problem. according to David Singmaster, Gardner says 1917)  (see June 21) [Since Kuratowski was 15 years old at this time, it could not be a proof of the houses and utilities problem, however it could have been proven by the Gem of Euler, (V - E + F = 2).  My version is here.  *PB ]

Kuratowski proved his theorem in 1930. Forty years later the dedication of Frank Harary’s classic Graph Theory was:
Who gave K5 and K3,3
To those who thought planarity
Was nothing but topology.
(In fact three other almost simultaneous discoveries of the theorem are recorded: Orrin Frink and Paul Althaus Smith; Lev Semenovich Pontrjagin; and Karl Menger!)  With thanks to *theoremoftheday ‏

*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

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