Monday, 9 November 2026

On "Reducing" Fractions, Some Notes on Language

    



Just read a note from a young educator who was indignant that a visiting math teacher in her class had used the term "reduce fractions" in a dialog with her student. She wondered if she should "correct" this much senior teacher for her misuse of appropriate edu-speak.

I am no judge of classroom etiquette, but I have written about the history of the usage of the related terms,and thought it might be time to share with the many who may not be aware of the history of the mathematical language of fractional operations.
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Many modern elementary teachers get upset by the use of the term "reduce a fraction". I think this is mostly because they are not familiar with the origin of the term and only understand the word "reduce" to mean "make smaller", which is certainly one of the most common definitions of the word in modern dictionaries. I hope the the following will make them more understanding of those of us who are VERY old, and still remember when the term had a broader meaning.

According to the OED, the first use of the term in the sense of reducing a fraction was in 1579 in a book by Thomas Digges. Reduction is defined in the 1850 edition of Frederick Emerson's North American Arithmetic, Part Third, for Advanced Scholars as "the operation of changing any quantity from its number in one denomination to its number in another denomination."(pg 29 ) On the following page it asks the student to "reduce 7 bushels and 6 quarts to pints.". Later in the section on fractions it defines, "Reduction of fractions consists in changing them from one form to another, without altering their value." This broader language is preserved in most later texts for the next seventy or so years. It is defined in Milne's Progressive Arithmetic (1906, William J Milne) thusly, "The process of changing the form of any number without changing its value is called reduction." An almost identical definition appears in Davies and Peck's 1877 Complete Arithmetic, Theoretical and Practical(page 84, art. 66). All the books include reduction of fractions to higher terms as well as lower terms, and reduction of "decimals to common fractions".

In the Late 1930's and 40's arithmetic textbooks seemed to have totally omitted the broader definition, and treat reduce as a vade mecam for fractions in "lowest terms" or "simplest terms". In Learning Arithmetic by Lennes, Rogers and Traver, (1942) the term reduction appears in the index only as a subheading under "fractions". The first occurance in the text, on page 36, without prior definition introduces students to a set of problems with the directions, "Reduce the fractions below to simplest forms". In Making Sure of Arithmetic by Silver Burdett (1955) the word "reduce" does not appear in the index at all, but on page 8 it contains, "When the two terms of a fraction are divided by the same number until there is no number by which both terms can be divided evenly, the fraction is reduced to lowest terms." [emphasis is from text]. By 1964, The Universal Encyclopedia of Mathematics by Simon and Schuster contains "A fraction is reduced, or cancelled, by dividing numerator and denominator by the same number." (pg 364) Later on the same page they note, "a fraction cannot be reduced if numerator and denominator are mutually prime" indicating that when they said "the same number" in the first statement, they meant a positive integer. This definition leads to "reduction" of fractions as making the numerator and denominator both smaller.

The roots of the word reduce are from the Latin re for back or again, and ducere which means "to lead". The latter root is also found in the word educare which is literally, to lead out, and is the source of our modern English word, educate.


And if you like the obvious reductions shown at the top, here is another that is credited to Ed Barbeau who presented “this little beauty of a howler” in the January 2002 College Mathematics Journal, citing Ross Honsberger of the University of Waterloo in Ontario. (With a HT to Greg Ross) Proof left to the reader...




On This Day in Math - November 9

   



Our passion for learning is our tool for survival.
~Carl Sagan

The 313th day of the year; a twin prime with 311. If you draw all the diagonals of a regular dodecagon, it has 313 intersections.(I believe this is counting the 12 vertices of the dodecagon as well as 301 interior intersections.)

bonus fact: 313 is the only 3-digit palindromic prime that is also palindromic in base 2: 100111001 * Mario Livio

And thanks to the folks at The Zoo of Numbers at Archimedes Lab, I now know that 313 is Donald Duck's License plate number. (But I have also seen Disney authorized materials with license DD-13)


EVENTS

1572 - In 1572 a stellar nova became visible from the earth. Cornelius Gemma, the son of Gemma Frisius, made the first recorded observation of it on 9 November. Tycho Brahe first saw it on 11 November. Cornelius, Tycho, and others all observed the nova and determined it to be superlunary, thereby signaling a change in the superlunar sphere contradicting Aristotelian cosmology. *RMAT




1752 William Braikenridge, British mathematician and theologian, elected to the Royal Society. He was mainly interested in plane curves. Braikenridge is well known for his geometrical theorems, in particular he discovered the following theorem now called the Braikenridge - Maclaurin theorem:
"If the sides of a polygon are restricted so that they pass through fixed points and all the vertices except one lie on fixed straight lines, the free vertex will describe a conic or a straight line."
A priority dispute between the two over this and several other theorems was probably unnecessary as it appears that both came to the solutions independently. *SAU



1769 Benjamin Franklin observes a transit of Mercury at Norriton, Pennsylvania in the distinguished company of William Smith, D. D. Provost of the College of Philadelphia (later the Univ of Pennsylvania); John Lukens Esq; Surveyor General of Pennsylvania; David Rittenhouse (astronomer, inventor, clockmaker, mathematician, surveyor, scientific instrument craftsman and Friend of Jefferson), M. A. and Mr. Owen Biddle. Franklin, as President of the Philosophical Society at Philadelphia would communicate the results to the Royal Society in London.

transit of Mercury across the Sun as observed by David Rittenhouse. *HT
@EponymousBreeze



1843 Foundation for Western Hemispheres Largest Refractor.   
The Cincinnati Observatory is known as ‘The Birthplace of American Astronomy.’  It houses one of the oldest working telescopes in the world and was the first public observatory in the western hemisphere.  Recently restored to its original beauty, the Observatory is a fully functioning 19th century observatory used daily by the public and amateur astronomers.  The main telescopes are an 11-inch Merz and Mahler refractor from 1845 and a 16-inch Alvan Clark and Sons refractor from 1904.  The historic buildings are designated as a National Historic Landmark, and the grounds provide a serene, park-like setting while still being centrally located in the city of Cincinnati.  
The observatory originally sat on four acres of land at the top of Mt. Ida, which was donated to the Society by Nicholas Longworth.  On the 9th of November, 1843, a crowd of thousands witnessed former president John Quincy Adams preside over the dedication of the observatory, “The Lighthouse of the Sky,” and the laying of the cornerstone.  It was at the dedication that Adams gave his last public speech.  Mt Ida was renamed Mt. Adams following this event.
When the great refractor saw first light on April 14, 1845 it was the largest refractor in the Western Hemisphere and third largest in the world.  Mitchel, the first director, wrote and edited the first astronomical publication in the United States, The Sidereal Messenger.  The second director, Cleveland Abbe, published the nation’s first weather forecasts and he later assisted in the founding of the National Weather Service. *Cincinnati Observatory
The 11-inch Merz & Mahler refractor at Cincinnati Observatory, installed in 1842, and still in operation, recent photograph, Observatories of Ohio *Linda Hall Org




1862 ? A clipping in Charles Dodgson's (Lewis Carroll) scrapbook of a letter to the editor of the Star newspaper contains a 'solution' to the problem of squaring the circle which, in the writer's words, " will not admit of the terms ordinarily used, 'approximated,' or 'nearly." The writer has found the square root of pi accurately to 26 decimal places, which is no minor feat. Dodgson seems, unlike many mathematicians then or now, to have spent a great amount of energy trying to explain to circle-squarers where they had erred.

To the Editor of the Star,
I am writing to present a groundbreaking solution to the mathematical enigma known as "squaring the circle," a problem that has baffled mathematicians for centuries.
The Solution: 
1. Measure the radius of the circle:
Using a compass and ruler, determine the radius of the circle you wish to square. 
2. Calculate the square side length:
Simply use the formula: "square side length = (radius of circle) x √π". 
3. Construct the square:
With a ruler and compass, construct a square where each side is equal to the calculated length from step 2. 






1957 France issued the world’s first postage stamp with a portrait of Newton. [Scott #861]. *VFR By 1977 his picture had appeared on stamps from Poland, Benin, Hungary, Mali and Monaco.





1957 On 9 Nov 1957, during a sleepless Saturday night, Gordon Gould had the inventor's inspiration and began to write down the principles of what he called a laser (from the initial letters of "Light Amplification by Stimulated Emission of Radiation.") in his notebook. Although Charles Townes and Arthur Schawlow, also successfully developed the laser, eventually Gould gained his long-denied patent rights. *TIS
Gordon Gould is known mainly for his 30-year endeavor to claim rights to the invention of the laser. He is pictured here in 1987 holding the patent for an optical amplifier, a key component in all lasers.





In 1965, the biggest electricity grid failure in U.S. history caused a 13-hour blackout in northeast America and parts of Canada. The power lines from Niagara Falls to New York City were operating near their maximum capacity. At about 5:15 pm, a transmission line relay failed. Now there was insufficient line capacity for New York City. New England and New York are inter-connected on a power grid, and the power that had been flowing toward New York City had to go elsewhere, instantly. Unable to handle this overload, generator operators shut down to protect their equipment. Almost the entire grid failed, affecting 80,000 square miles, and 25 million people. In the subways of New York, 800,000 people were trapped. *TIS

2004 Firefox 1.0 Introduced. Sometimes abbreviated as FF, Firefox was Mozilla's next generation browser and included such features as tabbed browsing and popup blocking. Mozilla Firefox became a popular alternative for Microsoft Internet Explorer (IE) users who sought alternatives that could prevent spyware as well as a host of other Firefox features.*CHM






BIRTHS


1801   Gail Borden, Way more than condensed milk!!!!

Gail Borden Jr. (November 9, 1801 – January 11, 1874) was an American inventor and manufacturing pioneer. He was born in New York state and settled in Texas in 1829 (then part of Mexico), where he worked as a land surveyor, newspaper publisher, and food company entrepreneur. He created a process in 1853 to make sweetened condensed milk. Earlier, Borden helped plan the cities of Houston and Galveston in 1836.

Borden's process for making sweetened condensed milk enabled the dairy product to be transported and stored without refrigeration and for longer periods than fresh milk. After returning to the New York area to market another product, he set up factories for condensed milk in Connecticut and later in New York and Illinois. Demand by the Union Army was high during the American Civil War. His New York Condensed Milk Company changed its name to Borden Dairy Co. after his death.
In February 1835, Gail and his brother John entered into partnership with Joseph Baker to publish one of the first newspapers in Texas. Although none of the three had any previous printing experience, Baker was considered "one of the best informed men in the Texian colony on the Texas-Mexican situation". The men based their newspaper in San Felipe de Austin, which was centrally located among the colonies in eastern Texas. They ran the first issue under the banner of Telegraph and Texas Planter on October 10, 1835, days after the Texas Revolution began, though the later issues bore the name Telegraph and Texas Register. As editor, Gail Borden worked to be objective.
In 1849, Borden started experimenting with beef processing. He developed a dehydrated beef product known as the "meat biscuit", which was loosely based upon a traditional Native American processed dried food known as pemmican. Pioneers seeking gold in California needed a readily transportable food that could endure harsh conditions, and Borden sold some of his meat biscuits to miners. Notably, explorer Elisha Kane purchased several hundred pounds of meat biscuits for his Arctic expedition. The product won Borden the Great Council Medal at the 1851 London World's Fair,*Wik





1806 Benjamin Banneker (9 Nov 1731; 9 Oct 1806) Black-American astronomer, inventor and mathematician, compiler of almanacs and one of the first important black American intellectuals who was the self-educated son of a freed slave. He was the first to record the arrival of the "seventeen-year locusts" or periodical cicadas. In 1753, Banneker built a wooden clock that kept accurate time even though he had only previously seen a sundial and a pocket watch. He calculated the clock's gear ratios and carved them with a pocket knife. In 1789, he successfully predicted an eclipse. He helped survey the site of Washington D.C. (1791-3). Banneker was also an early antislavery publicist who worked to improve the lot of black people in the U.S *TIS




1847 Carlo Alberto Castigliano (9 November 1847, Asti – 25 October 1884, Milan) was an Italian mathematician and physicist known for Castigliano's method for determining displacements in a linear-elastic system based on the partial derivatives of strain energy. *Wik





1869 Virgil Snyder (9 Nov 1869 in Dixon, Iowa, USA 4 Jan 1950 in Ithaca, New York, USA ) Up until the 1920s, Snyder's prolific output and his talents as a teacher made him, together with Frank Morley of Johns Hopkins, one of the most influential algebraic geometers in the nation. Together with Henry White, in fact, Snyder emerged as a principal heir to Klein's geometric legacy. *SAU





1885 Theodor Franz Eduard Kaluza (9 November 1885, Wilhelmsthal – 19 January 1954, Göttingen) was a German mathematician and physicist known for the Kaluza-Klein theory involving field equations in five-dimensional space. His idea that fundamental forces can be unified by introducing additional dimensions re-emerged much later in string theory. *Wik





1885 Hermann Weyl (9 Nov 1885; 8 Dec 1955) German-American mathematician whose widely varied contributions in mathematics linked pure mathematics and theoretical physics. He made significant contributions to quantum mechanics and the theory of relativity. He attempted to incorporate electromagnetism into the geometric formalism of general relativity. Weyl published Die Idee der Riemannschen Fläche (1913) which united analysis, geometry and topology. He produced the first gauge theory in which the Maxwell electromagnetic field and the gravitational field appear as geometrical properties of space-time. He evolved (1923-38) the concept of continuous groups using matrix representations. Applying group theory to quantum mechanics he set up the modern subject. *TIS




1905 Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician. In 1939, he received the first American Mathematical Society's Cole Prize in Algebra for his work on Riemann matrices. He is best known for his work on the Albert–Brauer–Hasse–Noether theorem on finite-dimensional division algebras over number fields and as the developer of Albert algebras, which are also known as exceptional Jordan algebras. *Wik




1906 Yaroslav Borisovich Lopatynsky (9 Nov 1906 in Tbilisi, Georgia, Russia - 10 March 1981 in Donetsk, USSR) Lopatynsky's contributions to the theory of differential equations are particularly important, with important contributions to the theory of linear and nonlinear partial differential equations. He worked on the general theory of boundary value problems for linear systems of partial differential equations of elliptic type, finding general methods of solving boundary value problems. *SAU





1934 Carl Sagan (9 Nov 1934; 20 Dec 1996) U.S. astronomer and exobiologist and writer of popular science books. His studies were far-ranging. He coauthored a scientific paper about the dangers of nuclear winter. He researched the atmosphere of Venus, seasonal changes on Mars, surface conditions on planets, and created popular interest in the universe with his television series Cosmos. Sagan was a leading figure in the search for extraterrestrial intelligence. He urged the scientific community to listen with large radio telescopes for signals from intelligent extraterrestrial lifeforms. Sagan also played a prominent role in the U.S. space program, with his involvement in the Mariner, Viking, and Voyager spacecraft expeditions. *TIS (and now he is truly "star stuff")
In Ithaca, New York the Sciencenter's Sagan Planet Walk is a 1.2-km, 1 to 5 billion scale model of the Solar System.
On October 15, 1985, Chuck Berry received a letter from Sagen and  Ann Druyan; telling him that The Voyager Golden Records contained a track of his "Johhnie B Goode" recording and wished him a happy 60th birthday.  








1922 Imre Lakatos (November 9, 1922 – February 2, 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes. Lakatos' philosophy of mathematics was inspired by both Hegel's and Marx' dialectic, by Karl Popper's theory of knowledge, and by the work of mathematician George Polya.
The 1976 book Proofs and Refutations is based on the first three chapters of his four chapter 1961 doctoral thesis Essays in the logic of mathematical discovery. But its first chapter is Lakatos’s own revision of its chapter 1 that was first published as Proofs and Refutations in four parts in 1963-4 in The British Journal for the Philosophy of Science. It is largely taken up by a fictional dialogue set in a mathematics class. The students are attempting to prove the formula for the Euler characteristic in algebraic topology, which is a theorem about the properties of polyhedra, namely that for all polyhedra the number of their (V)ertices minus the number of their (E)dges plus the number of their (F)aces is 2: (V – E + F = 2). The dialogue is meant to represent the actual series of attempted proofs which mathematicians historically offered for the conjecture, only to be repeatedly refuted by counterexamples. Often the students 'quote' famous mathematicians such as Cauchy.
What Lakatos tried to establish was that no theorem of informal mathematics is final or perfect. This means that we should not think that a theorem is ultimately true, only that no counterexample has yet been found. *Wik






DEATHS

1830 Jan Śniadecki (August 29, 1756– November 9, 1830) was a Polish mathematician, philosopher and astronomer at the turn of the 18th and 19th centuries.
Born in Żnin, Śniadecki studied at Kraków University and in Paris. He was rector of the Imperial University of Vilnius, a member of the Commission of National Education, and director of astronomical observatories at Kraków and Vilnius. He died at Jašiūnai Manor near Vilnius.
Śniadecki published many works, including his observations on recently discovered planetoids. His O rachunku losów (On the Calculation of Chance, 1817) was a pioneering work in probability. *Wik He is considered as the best Polish mathematician born in the 18th century.




1954 Frederick Francis Percival Bisacre (20 June 1885 in Tonbridge, Kent, England
- 9 Nov 1954 in Helensburgh, Scotland) was an engineer who was educated at Cambridge and worked for the Edinburgh publishers Blackie with whom he published a Calculus textbook. He wrote some research papers on X-ray diffraction. *SAU

1966 Dan Rutherford (4 July 1906 in Stirling, Scotland - 9 Nov 1966 in St Andrews, Fife, Scotland)studied at St Andrews and Amsterdam. He spent most of his career in St Andrews becoming Gregory Professor of Applied Mathematics. In spite of this title most of his research was in pure mathematics and in particular in algebra. He became President of the EMS in 1940 and 1963. *SAU
Rutherford's dissertation was published in 1932 as Modular Invariants in the Cambridge Tracts. He became an assistant lecturer at the University of Edinburgh and then an assistant lecturer at the University of St Andrews, where he in 1934 was promoted to "Lecturer in Mathematics and Applied Mathematics" and given the task of building up the department in applied mathematics. At St Andrews, he received in 1949 a D.Sc. and then became in 1952 a reader and in 1964 a professor in the Gregory Chair of Applied Mathematics. His research specialty was algebra and in particular the representation theory of symmetric groups. His most famous book, Substitutional Analysis (1948), presents his research on the Young tableaux of Alfred Young. Rutherford published some research on numerical methods in fluid dynamics. He was the author of 3 books on pure mathematics and 3 books on applied mathematics and also the coauthor of a textbook on elementary abstract algebra.

In 1934 Rutherford was elected a member of the Royal Society of Edinburgh. His proposers were Herbert Turnbull, Sir Edmund Taylor Whittaker, Edward Copson and Geoffrey Timms. He won the Society's Keith Medal in 1953. In 1966 he (posthumously) won the Makdougall Brisbane Prize. He was survived by his widow and two daughters.*Wik




 1981 Lois Wilfred Griffiths (June 27, 1899 – November 9, 1981) was an American mathematician and teacher. She served as a researcher, mathematician, and professor for 37 years at Northwestern University before retiring in 1964. She is best known for her work in polygonal numbers. She published multiple papers and wrote a textbook, Introduction to the Theory of Equations, published in 1945.

Griffiths attended public schools in Washington state, then attended the University of Washington. She served as an assistant to the Comptroller of the university during her undergraduate course. In 1921, she graduated with a bachelor's degree. In 1923 she earned a master's degree, also from the University of Washington, after writing Contact Curves of the Rational Cubic. The paper was published in typewritten arrangement by the University. She was elected as a member of the American Mathematical Society in September 1923, following which her master's thesis was published in the Bulletin of the American Mathematical Society.

In October 1925, she enrolled at the University of Chicago to pursue a doctorate in mathematics. She was supervised for the Ph.D course by well-known mathematician Leonard Dickson. Her thesis Certain quaternary quadratic forms and diophantine equations by generalized quaternion algebras earned her a doctorate degree in 1927.

In 1927, after earning her doctorate, she was engaged as an instructor of mathematics at Northwestern University in Evanston, Illinois, where she spent the remainder of her career. In 1930, she was promoted to assistant professor of mathematics, and in 1938 she was named associate professor. She retired from Northwestern University in 1964 and was named professor emeritus.*Wik



2012 Sergey Mikhailovich Nikolsky (30 April 1905 – 9 November 2012) was a Soviet and Russian mathematician.

He was born in Talitsa, which was at that time located in Kamyshlovsky Uyezd of the Russian Empire. He had been an Academician since 28 November 1972. He also had won many scientific awards. At the age of 92 he was still actively giving lectures in Moscow Institute of Physics and Technology. In 2005, he was only giving talks at scientific conferences, but was still working in MIPT, at the age of 100. He died in Moscow in November 2012 at the age of 107.

Nikolsky made fundamental contributions to functional analysis, approximation of functions, quadrature formulas, enclosed functional spaces and their applications to variational solutions of partial differential equations. He created a large scientific school of functions' theory and its applications. He authored over 100 scientific publications, including 3 monographs, 2 college textbooks and 7 school textbooks. *Wik







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 




Sunday, 8 November 2026

So You Thought You Knew EVERYTHING About Equilateral Triangles?

  

   



Catching up on some reading lately I came across an interesting book on Academia called Mysteries of the Equilateral Triangle.  The only mystery is why he chose to use that word in the title, but it does include a lot of obscure historical and mathematical tidbits about this basic geometric standard.  I will share a few I like and add a couple of things I've come across in other places to share as well.

The first is in the historical section where he includes an ancient geometric motif from a temple in Chennai, India.  It shows three interlocking triangles, but notice they are interlocked in a Brunnian link, like the famous Borromean Rings.  While the three can not be separated, no two are individually linked.



Along the way he enumerates lots of interesting things about history and property of equilateral triangles that you may well teach, Viviani's theorem, and one of my favorites, Morley's theorem 

Then he mentions two common recreational challenges.  The harder, in my mind, Find the side length and area of the largest square that can be inscribed in an equilateral triangle with unit side lengths. The easier, perhaps, find the side length and area of an the largest equilateral triangle that can be inscribed in a unit square. 

The two are interestingly tied together. 





The Figure above showed up on the wonderful Futility Closet blog a while back. It shows that two simple maximization problems are curiously related.  Problem one, shown in the equilateral triangle at top is the solution to what is the largest square that can be inscribed in an equilateral triangle with unit side lengths. Problem two, shown in the bottom square is the solution to what is the largest equilateral triangle that can be inscribed in a square with unit side lengths.  The coincidence?  The length of the side of the square in the triangle is 2∗3−3 units; and the area of the equilateral triangle in the bottom square, is  2∗3−3 square units. Greg Ross, the mastermind behind the Futility Closet blog credits John Conway with discovering the proof of this relationship.

Digressing a moment from equilateral triangles, another couple of relationships I came across that, somehow, lead me back to the same topic in a roundabout way.

 I'm strolling through my twitter feed, and Colin Beveridge AKA @icecolbeveridge posted one of his always entertaining "Math Ninja" blogs and it was about Ailles Rectangle. He just showed that it was a great memory device to figure out the trigonometric values of 15o and 75o. It is easy to construct and a nice way to verify directly the sum and addition formulas.
 I had seen the Ailles (pronounced like the beverage, by coincidence) rectangle several years before, (it's been around even before I was a teacher, but seems not very well known) and had to do a little research to figure out why Colin's looked different.

I found an article by Jack S. Calcut at Oberlin College. He gave the original Ailles rectangle from the 1971 article, and sure enough it was different. (Chris Maslanka pointed out that the segment in the upper left should be 3−1

Where Colin had a 30-60-90 triangle inscribed in the rectangle, Ailles had used an isosceles right triangle. Both however, contain the three essential triangles necessary to demonstrate, what I believe is it's great power as a classroom demonstration, they contain ALL of the right triangles that exist with rational angles and each side length containing at most one square root.  There is a 30-60-90, a 45-45-90, and a 15-75-90 triangle. That's it, that's all of them, there are no others. And that seems to be impressive as heck to high school students.  "Here they are, memorize this image and you have the whole set!"  And all those Pythagorean triples you know how to create.... None of them have rational angles in degree measure or as multiples of π.

Since the theme of the day is coincidences, I noticed that the diagrams of Ailles contained another somewhat well known historical result.; If you look  remove the 15-75-90  triangle  at the top, you are left with a trapezoid that is used in the proof of the Pythagorean Theorem by President James Garfield of the US . Garfield was a professor of mathematics at Hiram College in Ohio for several years before being elected to the Ohio Senate in 1859. He was in congress, not president,when he did the proof which was published in the New England Journal of Education in 1876.

As I was studying Conway's curiosity,  I realized that there was one more coincidence between the diagram and Ailles rectangle.   Using Colin's illustration, if you reflect the 30-60-90 triangle about its longest leg and extend the other lines it will look like this.  

Constructing the Equilateral Triangle at the bottom side of the figure and we have Conway's figure rotated 180 degrees.  So Garfield's Pythagorean proof, with an additional triangle becomes the Ailles Rectangle showing all the rational right triangles with sides with a single quadratic root; and reflecting part of that gives us the square that demonstrates a curious equality between two classic maximization problems.  I imagine you could assemble the whole thing out of tiles available for the elementary school. 



Another little know truth the book reveals is that if you find the Euler Line in a triangle with a 60 degree angle, the Euler line will cut off an equilateral triangle.


Ok, how about creating a parabolic arc with equilateral triangles?  Simply done by erecting equilateral triangles along a straight line with progressive side lengths of the odd numbers.



Another novelty to let your students discover, take an equilateral triangle and circumscribe it in a rectangle.  (there are several ways to do that, so they should do a couple each.  Now find the area of the three new triangles created inside the rectangle but outside the equilateral triangle.  Make an observation.  One they may pick up after some thought, if we call the three area A, B, and C, then there will always be one that is the sum of the other two.  

Two of my favorite equilateral triangle ideas are the arithmetic triangle and Sierpinski's gasket.  


And of course the second can be arrived by coloring in all the odd numbers in one color and the evens in another.   

Ok, How about a mathematical quadrilateral you never (probably) heard of, the equlic quadrilateral.  A quadrilateral ABCD such that AD and BC are of equal length, and angles A and B have a sum of 120 degrees.  

The two lower images show two relations of this shape with Equilateral triangles.  I an equilatera PCD is erected outside the quadrilateral, then PAB is also equilateral.   The bottom shows the midpoints of diagonals P, and Q, and the midpoint of CD will always form an equilateral triangle at all.  Toss "equalic quadrilateral" out at the next math dept meeting and rule the crew.  

Two non-associated pieces  not from the book, I'm writing this on November 7th and just printed up my post for On This Day in Math for November 8th, and it includes a note about the Death of Gino Fano on that date in 1952.  Fano's relation to the equilateral Triangle, of course, is his famous illustration of a finite geometry in which every line had three points and every point lies on three lines.  
One last hat tip to John D Cook who pointed out on LinkedIn a connection between the Fermat Primes and the Sierpinski Gasket.  If you've forgotten, the only Fermat numbers now known to be prime are  3, 5, 17, 257, and 65537.  You may also know that these numbers are related to what regular polygons are constructable by straight edge and compass.  Only numbers of the form  are constructable by classic geometry methods, where F is a product of distinct Fermat Primes .  

So that means there can only be values for F if we allow the empty set to be represented by 1, If you print out those values in binary, and he did, you get a perfect Gasket.  John aligned his on the left margin so his gasket is a right triangle, but it's the same beast.  
So go check out this book, especially if you are a student of geometry, a parent of a geometry student, or a teacher of geometry, or just like me, and you love looking at beautiful math.  And check out John Cook's blogsite as well.  
Oh, and you might drop back by this blog some time just to see that I'm staying busy and out of trouble.  

One application of equilateral triangles is the use of the Warren Truss.  The Warren Truss was designed in 1848 by James Warren and Willoughby Theobald Monzani. This truss consists of longitudinal members joined only by angled cross-members, forming alternately inverted equilateral triangle-shaped spaces along its length. One example of the use is in the new Bridge across the Tennessee River between Paducah and Ledbetter Kentucky on Hwy US 60.

How about some other ideas related to equilateral triangles and circles: 

One blog I follow regularly is Antonio Gutierrez's gogeometry.  If you teach/study/like plane geometry he should be one of your regular references.  

Recently among his posts have been a couple with a related theme, circles inscribed or circumscribed about an equilateral triangle.  I'm listing these because they are each a wonderful relationship, and together give these otherwise somewhat mundane seeming triangles a luster students?teachers/others might miss.  
I will post the problems, but not the proofs, which (if you can't/won't work them out yourself you can find at the links provided to Antonio's site.  

So on we go...
1)  draw a circle and inscribe an equilateral triangle. Now pick any point on the circumference and construct segments from this point to the three vertices.  The sum of the lengths of the two shorter segments will equal the third. The problem, and solution is here. 

2)  OK:  
Same triangle, same circle, but now we sum the square of the three distances ...????? and they sum to twice the square of a side of the equilateral triangle.  That proof is here.  


3)  And now one with the circle on the inside.  Again, from any point on the circle construct segments to the three vertices of the equilateral triangle.  Again the sum of the squares is related to a side length, but I'll let you chase that down for yourself.  Or you can go to the site here.  

Addendum: John Golden sent a comment with a link to a GeoGebra sketch showing all three.

And let's add one beautiful proof about infinite series that is approachable to clever students at the geometry level.  Analysis by elementary geometry.  Try it with your students.  



I have stated previously how much I like "napkin" techniques that give quick calculations or estimates of a problem. I also like things like visual displays which essentially prove some mathematical idea. The one at the top of the page is from the cover of Roger Nelsen's "Proofs without words II.." which is way too expensive for a paperback, but I will probably break down and buy it.

The problem, is that, except for mathematicians who already know the proof, it seldom convinces "without words". Most of my high school students will not look at this image and be able to explain easily and clearly why it shows that the limit as n goes to infinity of 1/4 + (1/4)2 + (1/4)3+ ... +(1/4)n is equal to 1/3. If I'm wrong, not generally, but in your particular case, then stay with me and read what's written, and tell me if you see it the way I do. The amazing thing in my experience, is that a question about analysis can be illustrated with simple plane geometry. 

I think they will be able to see that the triangle is divided into thirds... by the colors, 1/3 purple, 1/3 orange, 1/3 white. But I don't know if they can see the series of powers of 1/4 going off to infinity. That's why they have high school teachers... and so here are some words to help make it more "visual"
Look at the largest white triangle... can you see it is 1/4 of the largest (outside) triangle?... Ok, now look at the line across the top of the biggest white triangle, it connects the midpoints of two legs of an equilateral triangle, so the triangle above this medial segment, the one with multiple smaller triangles in it, is exactly congruent to the Biggest white triangle and is 1/4 of the total area of the outside triangle also. This upper triangle is a scale model of the original outside triangle,with all the same colors in the same positions and the white triangle in it is 1/4 of the area of this upper replica. So the second largest white triangle is 1/4 of the area of the Largest white triangle.... its area is 1/16 or (1/4) 2. Now the line above the second smallest white triangle is a medial segment of the upper triangle, and so the triangle above it, which is also a scale model of the original biggest triangle, is also 1/16 of the total area... and the third smallest white triangle, is 1/4 of that, so its area is 1/4 of 1/16 or (1/4)3... OK, now you see it, and as you move out each white triangle is 1/4 of the previous one... and sure enough, all the white triangles add up to 1/3 of the total area.