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Benford's Law If you looked in lots of reference books and found the areas of all the lakes on the Earth, about 30% of the numbers you would find would start with a 1. It doesn't even matter if some of the books gave area in square miles, others in hectares, and still others in square meters. This is one of the surprising results of Benfords Law.

The same result would occur if you found the daily sales for all the Macdonald's franchises in the world, and again, it doesn't matter that some are in dollars and others in yen. This result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude. The law is named for US Physicist Frank Benford who published a description of the effect in 1938.

As you might have guessed, someone else did it earlier; a half century earlier. In 1881 a note to the American Journal of Mathematics by an American astronomer named Simon Newcomb described an unusual observation. He had noticed that the tables of logarithms that were in common use back then by astronomers, always had the pages of the lower numbers more dog-eared than the pages of the higher numbers. He suggested that natural observations tend to start with the number one more often than with an eight or nine. For some reason, the observation went without much comment. Years later Benford published data from an assortment of different areas, and the mathematical quirk of nature now bears his name. No reason was given for the unusual distribution until 1996, when Theodore Hill of the Georgia Institute of Technology published, what else, Hill's Theorem.

There is even a formula for how the distribution works. According to Benford's Law, If you take a large data sample from a collection of non-random but wide ranging data, the percentage starting with the digit d will be \( log(1 + \frac {1}{d})\) where the base ten log is intended. Since log(2) \( \approx .301\) we estimate that 30% of the numbers you will see on the internet in a day start with 1. \( log(1 + \frac{1}{2})\approx .176 \) so 17.6% start with two and each larger digit becomes a little less common with a little less than 5% of all numbers starting with nine.

The applications of Benford's Law are just starting to emerge in the area of detecting fraud. Several cases have already been found. The New Scientist reported that over a million dollars in fraud was discovered using this process in a healthcare incident. The computer search showed an unusually large distribution of claims beginning with a six, and a large number of them turned out to be bogus. Bunko artists of the future will have to be better mathematicians, it seems

This web page shows how several different data sets demonstrate Benford's Law.

Newcomb's paper is available digitally. Simon Newcomb (1881). "Note on the frequency of use of the different digits in natural numbers". American Journal of Mathematics (American Journal of Mathematics, Vol. 4, No. 1) 4 (1/4): 39–40.



Brahmagupta's Formula The Indian Mathematician Brahmagupta (598-670) is often claimed to be the greatest mathematician of the dark ages. Whether he was the greatest or not, he did make great strides in working with zero including writing out rules for mathematical operations with zero. His writing also provides the earliest example I know of a multiplication system very similar to the common algorithm taught today.  One of his many achievements was to expand the method we now know as Heron's formula (or Hero's formula) to cyclic quadrilaterals.   A cyclic quadrilateral is a quadrilateral that has all four vertices on a circle.     The formula of Brahmagupta extends the Heron method by reducing the semi-perimeter, s, by each of the four sides.  In this way Heron's formula can be thought of as a special case of a cyclic quadrilateral when one side is diminished to zero to form a triangle.
  The s in Brahmagupta's method, like Heron's, is one half the perimeter of the figure, s = (a+b+c+d)/2.  The formula for the area of a cyclic quadrilateral, \( area= \sqrt{(s)(s-a)(s-b)(s-c)} \), can be extended to apply to all quadrilaterals by adding another term under the radical, \( area= \sqrt{(s)(s-a)(s-b)(s-c)-(abcd)Cos \frac{\theta_1 + \theta_2}{2}} \).  The two angles can be either pair of opposite angles.  In a cyclic quadrilateral each pair of opposite angles sum to \(\pi\) so that this final term reduces to zero.



Butts and Bounds I had never heard these two terms used as math terms until I came across "The Word Museum: The Most Remarkable English Words Ever Forgotten" by Jeffrey Kacirk. He writes:

Butts and Bounds; words used in describing the boundaries of land. Properly speaking, butts are the lines at the ends, and bounds are those on the sides, if the land is rectangular shaped. But in irregular shaped land, butts are points or corners where the boundary lines change directions.
So for rectangular area, A = BxB. The word may not be as forgotten as the book title suggests. A quick search revealed that the term appears in a law dictionary as late as 1998 with the note, "BUTTS AND BOUNDS. A phrase sometimes used in conveyancing,where a particular piece of land is described ..."... (and just a personal footnote here, don't have the kids in the room when you do an internet search of "butts")


Carnot's Theorem is named for Lazare Carnot, the father of Sadi Carnot who is remembered for his work in thermodynamics. The senior Carnot is best remembered for his geometric works including De la correlation des figures de géométrie , an 1801 publication in which he tried to generalize many of Euclid's theorems.

His discoveries include an unexpected relationship between the radii of the inscribed and circumscribed circles of a triangle. The theorem states that the sum of the distances from the circumcenter, O, to the three sides is equal to the sum of the radii of the incircle and the circumcircle. In the figure the sum of the three dotted lines is equal to r + R, the radii of the two circles. If the triangle is obtuse, the distance from the circumcenter to the nearest side is treated as negative, as this length lies outside the triangle. A proof of the theorem can be found at the Cut-the-Knot web page.



Cybernetics is the theoretical study of control processes in fields such as electronics or biology. The term was aptly coined by Norbert Wiener from the Greek word kubernetes, for a helmsman or pilot. The root is kubernan for guide or steer. Here is a link for further information on the history and development of cybernetics



Klein Bottle The Klein bottle is named for German mathematician Felix Klein, but I can find nothing that indicates he thought it up. Unfortunately, I can not find anything that says who did. The (more or less) technical definition of a Klein bottle is that it is a closed non-orientable surface with Euler Characteristic equal to zero. OK, maybe you should get the explanation from the people at Ohio State Univ They show you how it can be imagined starting from a single flat sheet of paper.

The image at right is one of two types of Kline bottles possible. The second type is called the figure eight model. The picture and its companion can be seen at the Geometry Center. If you've seen enough pictures and want to see a real Klein bottle (OK it is impossible) and maybe buy one of your very own see the page of this guy selling nice fake Klein bottles.

If games are more your line, check out this Key Curriculum site where you can play tic-tac-toe on a Klein bottle Not your style? Then perhaps you want to drag out the lego set and make a Lego Kline bottle like the one Andrew Lispon made on this link.

If none of that is what you came looking for, then maybe it is the limerick, author unknown that brought you to my page.... so here it is:

A mathematician named Klein
Thought the Mobius strip was divine
He said "If you glue
The edges of two
You can make a strange bottle like mine".



Mobius Strip Perhaps one of the best known mathematical objects outside the basic geometric shapes, the Mobius strip is named for August F. Mobius. Mobius had been working on the problem of finding a classification system for generalizing Euler's formula for polyhedra, V + F - E = 2. While exploring the shapes that could be made with a strip of equilateral triangles, he realized that if the strip was twisted and its ends joined, it would form a surface which effectively had only a single side. Strangely it seems that only a few months earlier (July of 1858), another mathematician working on the Euler formula, J. B. Listing, also came across the idea of a one-sided figure. Listing not only discovered the idea first, he also published his idea first in 1861. (I later learned that Listing is also the origintor of the term topology, "The word TOPOLOGY was introduced in German in 1847 by Johann Benedict Listing (1808-1882) in "Vorstudien zur Topologie," Vandenhoeck und Ruprecht, Göttingen, pp. 67, 1848. However, Listing had already used the word for ten years in correspondence. ") The paper which outlined Mobius' investigations was not published until 1869, a year after his death.

The stip is not without some practical application. For years farmers and saw mill operators knew that putting Mobius twist in a flexible drive belt would keep it from wearing unevenly. I even read that Goodyear has patented a type of conveyor belt that is a Mobius strip for the same reason.

One of the most famous images of a Mobius strip is in this picture by Dutch artist M C Escher. See more of Escher's work at the sight where I got this image. Here is a link to down load a lesson plan with instructions and some explorations of the properties. By 1881 the properties of the one-sided figure were being used in a magic trick called the Afghan band. How to do the Afghan band and several other clever "magic" feats can be found at this sight. Another nice link for teachers and students is this article by Ivars Peterson about the relationship between Mobius bands and recycling. Peterson also has written another article about some three-dimensional versions of the one-sided band developed by artists and architechts. One of the famous examples is a sculpture by Max Bill, who had never heard of Mobius or his strip. The one shown here is dated about 1930, although he continued to make many versions of continuous strips for decades. Another famous architect/scientist who was attracted to 3-D variations of the Mobius strip was Robert R Wilson, the first head of the Fermi Laboratory in Batavia, Illinois.

The famous one sided band even made its way into classic humor. I found a Mobius Strip/Chicken Joke at the Numericana site of Gérard P. Michon. which also includes lots of great math.

Q: Why did the chicken cross the Möbius strip?
A: To get to the same side.

On the other hand, the famous twisted band is the subject of poetry also, to wit:

A mathematician confided
That a Moebius band is one-sided
And you'll get quite a laugh
If you cut one in half
For it stays in one piece when divided
-------------author unknown..



Negative (photographic) The photographic negative seems to have been created by William Henry Fox Talbot of the rural village of Lacock in Wiltshire, England around 1830. Fans of Harry Potter movies (at the time I am writing this a second is being made) may be interested to know that Talbot owned and lived in the Lacock Abbey which was the backdrop for the school in the movies. A while after writing the above I discovered another note about Fox Talbot:

>I am looking for the original source of the following theorem:
Four lines determine a complete quadrilateral.
The midpoints of its diagonals lie on a line - call this the gauss line.
Then five given lines will determine five gauss lines. It turns out that these are concurrent.
This result (it's stunning with Cabri) appears as an exercise in Durell's "Projective geometry" (p199, ex 16) and Casey's "Sequel to Euclid" (p159, ex 101) where it is ascribed to H Fox Talbot.
Now William Henry Fox Talbot (1800-1877) was a famous early pioneer of photography. He read mathematics at Cambridge and was an enthusiastic amateur who wrote a number of papers published mainly in the Philosophical Transactions of the Royal Society. I have not been able to trace any paper which announces the above theorem, and I wondered whether anyone has an easy way of finding this out.
Dick Tahta (d.tahta@open.ac.uk)
An answer from Julio Gonzalez Cabillon confirmed that it was the same Fox Talbot
Jakob Steiner credits William Henry Talbot in a quotation appeared in the GERGONNE's journal [after Joseph D. Gergonne (1771-1859), founder and editor of the *Annales de mathematiques pures et appliquees*, Paris, L'Imprimerie de Durand-Belle, 1810/11-1831/32 --the first periodical dedicated *exclusively* to mathematics].

Cf. Annales de Gergonne, tome XVIII, p. 302, 1827-1828.

Q U E S T I O N S P R O P O S E E S.
Theoreme sur le quadrilatere complet;
Propose a demontrer par M. J. Steiner, geometre, de Berlin (*).

_____________________________________________________________
(*) [...] Nous saisirons donc, avec plaisir, cette occasion de declarer que l'elegant theoreme demontre par M. Lentheric, a la page 366 de notre XVIIe volume, nous a ete indique par M. W. H. Talbot, de la Societe philosophique de Cambridge.

It is now thought that the foundation of photography, the idea that some chemicals containing silver would turn black when exposed to light, was known as far back as the 13th century, and was known to Saint Albertus (then called Albertus Magnus or Albert the Great) while he was a teacher in Paris. By the early 19th century small darkened enclosures called Camera Obscura were used to make copies of outside images. A small pinhole was made to allow light to shine in and project an inverted image on a white surface allowing the image to be traced or painted. It is thought that the Chinese may have understood the projective lens quality of a pinhole as far back as 400 BC. Our word camera is a shortened version of camera obscura. The Latin meaning of camera was a chamber or room, and the obscura was for dark, so the literal meaning of camera obscura was "dark room" or "dark place". The old meaning of camera persists in legal language still. The legal term "in camera" means in private or in the judge's chambers. It was in his camera obscur that Talbot experimented with placing a silvered paper in the projection area of the camera and exposing it for a short time, then arresting the process by dipping the paper in a salt solution to "fix" the image. This was similar to the method used at about the same time by Daguerre in France, except that Daguerre used an intermediate step of exposing the silver to a mercury vapor that reversed the color of the silver nitrate giving a "positive" image, and also destroying the health of the photographer who developed the film.

The process of "fixing" the film with salt solution was discovered by John Herschel, son of the famous astronomer. He is also credited with the creation of the words "photography" and "negative" (for the photographic negative), both in 1839. He also seems to have been the first to use the term "snapshot".

Photograhy combines the Greek roots photos for light, with graphos to draw or mark. Here is a link to the mathematical term negative.



Oblique The word oblique (sometimes oblike) is less commonly used today than in the past, but still emerges enough that it should be part of every math student's vocabulary. Oblique is used to refer to something other than a right angle. Triangles withou a right triangle are rightfully called oblique triangles. Cones in which the altitude does not intercept the center of the base are also referred to as Oblique. Two intersecting lines that are neither parallel or perpendicular are also called oblique. There is even a military marching maneuver that directs the group to change direction at a 45o angle that is called an "oblique march". The word is from the Latin obliquus combining ob (to or toward) + liquis (bent upward). The common usage is related to something that is slanted or inclined.



Ogive The statistical term ogive (which is also called an ogee) refers to a cumulative distribution function.


The term was applied by Francis Galton to the cumulative normal distribution but is used more generally now to apply to almost any cumulative distribution. [1875 F. GALTON in Philos. Mag. 49 35 "When the objects are marshalled in the order of their magnitude along a level base at equal distances apart, a line drawn freely through the tops of the ordinates..will form a curve of double curvature... Such a curve is called, in the phraseology of architects, an ‘ogive’ ".... OED] Ron Dirkse, a math teacher from the American School in Japan recently posted, "In Stigler's History of Statistics --The Measurement of Uncertainty before 1800 (ISBN 0-674-40341), the section on the Englishman, Francis Galton: page 268, it states, ' Borrowing a term from architecture, Galton called this curve an 'ogive'; we now call it the inverse normal cumulative distribution function'." Here is a copy of the illustration from Galton's article (or a facsimile thereof).

The ogee was a common decorative element in many of the English Churches around 1400. The image below shows the door to the Church of The Holy Cross at Caston in Norfolk. In this image you can see the use of the ogee in the design of the door and repeated in the windows above.

Both ogive and ogeeThe word originally come from a term in architecture for a diagonal rib of a Gothic vault or a pointed arch. The root is the Late Latin obvita, the feminine past participle of obvire, to resist or go against (and thus supporting). Obviate is from the same root.

I have recently seen the term (sometimes with the alternative spelling ogee) used in modern geography (applied to certain galciers) and in Physics/aeronautics.

Cumulative distributions existed before Galton's application of the term, of course, and one example credited to Fourier in 1821 appears here. The graph shows the number of people in Paris by age (the number per 10,000 in 1817.who were of a given age or over).



Pareto's Principle The Pareto Principle states that most of the result is attributable to a few of the causes. Examples often stated include items such as "80% of the accidents involve only 20% of the drivers. The principle is named for Wilfredo Pareto , a nineteenth century Italian Economist. The actual principal was first stated by Joseph Juran, for application to the field of quality management and named in honor of Pareto. In quality management and some other areas of statistical use Pareto's name is also applied to a bar chart ordered so that the events are in order from most frequent to least frequent. The figure shows an example of a Pareto chart I created by using Fathom Software. The vertical scale is sometimes used to represent the costs of outcomes rather than the actual frequency.



Pigeonhole Principle The basic idea behind this mathematical principle is what students would call common sense; if there are n objects to be placed in m receptacles (with m less than n), at least two of the items must go into the same container. While the idea is common sense, in the hands of a capable mathematician it can be made to do uncommon things. Here is a link to an article by Alexander Bogomolny in which he uses the principle to argue that there must be at least two persons in New York City with the same number of hairs on their head.

The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. The pigeon seems to be a recent addition, as Jeff Miller's web site on the first use of some math words gives, "Pigeon-hole principle occurs in English in Paul Erdös and R. Rado, "A partition calculus in set theory," Bull. Am. Math. Soc. 62 (Sept. 1956)". In a recent discussion on a history group Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portugese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish we use also:"the principle of the drawers of Dirichlet" that is 'Zasada szufladkowa Dirichleta' ". Dirichlet first wrote about it in " Recherches sur les formes quadratiques ŕ coefficients et ŕ indéterminées complexes" (J. reine u. angew. Math. (24 (1842) 291 371) = Math. Werke, (1889 1897), which was reprinted by Chelsea, 1969, vol. I, pp. 533 618. On pp. 579 580, he uses the principle to find good rational approximations. He doesn't give it a name. In later works he called it the "Schubfach Prinzip" [which I am told means "drawer principle" in German]

I had assumed, as stated on the Wolfram "MathWorld" site, that,"This statement has important applications in number theory and was first stated by Dirichlet in 1834". In truth, the principal has been around much longer than Dirichlet, as I found out in June of 2009 when Dave Renfro sent me word that the idea pops up in the unexpected (at least by me) work, "Portraits of the seventeenth century, historic and literary", by Charles Augustin Sainte-Beuve. During his description of Mme. de Longuevillle, who was Ann-Genevieve De Bourbon, and lived from 1619 to 1679 he tells the following story:
"I asked M. Nicole (See below for description of M. Nicole) one day what was the character of Mme. de Longueville's mind; he told me she had a very keen and very delicate mind in knowledge of the character of individuals, but that it was very small, very weak, very limited on matters of science and reasoning, and on all speculative matters in which there was no question of sentiment ' For example,' added he, ' I told her one day that I could bet and prove that there were in Paris at least two inhabitants who had the same number of hairs upon their head, though I could not point out who were those two persons. She said i could not be certain of it until I had counted the hairs of the two persons. Here is my demonstration/ I said to her: M lay it down as a fact that the best-fiimbhed head does not possess more than 2oo,ooo hairs, and the most scantily furnished head b that which has only 1 hair. If, now you suppose that 2oo,ooo heads all have a different number of hairs, they must each have one of the numbers of hairs which are between i and 2oo,ooo; for if we suppose that there were 2 among these 200,000 who had the same number of hairs, I win my bet But suppose these 200,000 inhabitants all have a different number of hairs, if I bring in a single other inhabitant who has hairs and has no more than 200,000 of them, it necessarily follows that this number of hairs, whatever it b, will be found between 1 and 200,000, and, consequently, b equal in number of hairs to one of the 200,000 heads. Now, as instead of one inhabitant more than 200,000, there are, in all, nearly 800,000 inhabitants in Paris, you see plainly that there must be many heads equal in number of hairs, although I have not counted them.' Mme. de Longuevillle still could not understand that demonstration could be made of the equality in number of hairs, and she always maintained that the only way to prove it was to count them. "
The M. Nicole who demonstrated the principal was Pierre Nicole, (1625 -1695), one of the most distinguished of the French Jansenists writers, sometimes compared more favorably than Pascal for his writings on the moral reasoning of the Port Royal Jansenists. It may be that he had picked up the principal from Antoine Arnauld, another Port Royal Jansensist who was an influential mathematician and logician. Here is a segment from his bio at the St. Andrews Math History site.
------------------------- He published Port-Royal Grammar in 1660 which was strongly influenced by Descartes' Regulae. In Port-Royal Grammar Arnauld argued that mental processes and grammar are virtually the same thing. Since mental processes are carried out by all human beings, he argued for a universal grammar. Modern linguistic theorists consider this work as the beginnings of the modern approach their subject. Arnauld's next work was Port-Royal Logic which was another book of major importance. It was also strongly influenced by Descartes' Regulae and also gave a first hand account of Pascal's Méthode. This work presented a theory of ideas which remained important in philosophy courses until comparatively recent times. In 1667 Arnauld published New Elements of Geometry. This work was based on Euclid's Elements and was intended to give a new approach to teaching geometry rather than new geometrical theorems."
He was a corresopondent of Gottfried Wilhelm Leibniz, and of course Pascal, who wrote the Pascal "Provincial Letters" in support of Arnauld. I enjoyed the quote about him from the Wikipedia bio: "His inexhaustible energy is best expressed by his famous reply to Nicole, who complained of feeling tired. 'Tired!' echoed Arnauld, 'when you have all eternity to rest in?"

I have not been able to find any thing in Arnauld's personal writing at this time to confirm that he was aware of or used the Pigeon-hole Principle. I have also seen a comment that there is a book by Henry (or Henrik) van Etten (pseudonym of Jean Leurechon, who coined the term thermometer) , circa 1624, which uses the method for problems involving "if there are more pages than words on any page" and various other illustrations. The writer suggests that the problem is in the French version but not the English translation. Would love to hear from someone who can confirm, and perhaps send a digital image.



Quaternions In the first decade of the 19th century several mathematicians were developing a framework for what would become vector analysis. In 1813 J. Francais published a short piece in Gergonne's Annales in which he described a geometrical view of complex numbers which had come to him from letters in the estate of his deceased brother from Legendre. In the letters Legendre had described the ideas as the work of a friend. The friend in question, John Argand, announced himself to the publication summarizing his previously published book, and describing his unsuccessful efforts to extend the work to three dimensions. In a criticism of Argand's 3-d attempts, Servois responded with his own methods of spatial analysis. In his notes he described six values that would be needed to satisfy his spatial geometry. Years later Hamilton would call this a near approach to the quaternions.

On October 16th, 1843, Sir William Rowan Hamilton had an inspired thought while walking with Lady Hamilton along the Dublin Royal Canal beside the Bourgham Bridge. At some time he carved the identities that formed the basis of quaternions in the stone wall of the bridge. The image at right as well as more information about the annual canal walk, and more pictures of the historical canal can be seen at this web site.

The quaternions can be described as a four dimensional vector with one scalar and three imaginary parts. The imaginary identities are often labeled i, j, and k so that a quaternion, q, can be expressed as a + bi + cj + dk where a,b,c,d are real coefficients and i, j, and k are vector quantities such that i2 = j2 = k2 = ijk = -1 and ij=k, jk=i, ki=j and ji=-k, kj=-i, ik=-j. These six quantities, i, j, k, -i, -j, and -k, were to Hamilton, the six quantities that Servois had named, but not found.

The fact that ij is not equal to ji make the multiplication of quaternions one of the first non-commutative operations described in mathematics. If the ordinary rules of distributive multiplication are applied to the multiplication of quaternions q and r, with zero scalar terms, the result will be a quaternion that has a scalar that is the negative of the dot or scalar product of the coefficients, and the vector portion will be the cross or vector product of the vector portions of the two factors.



Sample The word sample is such a common part of statistical language today, that it seems hard to imagine that it only came into common usage in the last one hundred years. In the syllabus for the 1885 lectures of F Y Edgeworth at King's College in London he uses "... if a considerable set, say a thousand men, is taken at random..." The absence of the use of the word sample in such usage almost assures that the common usage began later than these papers were written. The 1913 Webster's Unabridged dictionary includes the definition "To take or to test a sample or samples of; as, to sample sugar, teas, wools, cloth" which sounds very like the application of the statistical idea. Jeff Miller's web page on the first use of mathematical terms does not mention sample itself, but on the term "Sample Space" he supplies a date of 1938 with mention of a possible earlier, but unspecified date of usage. The usage in statistics is essentially unchanged from its common usage as a representative subset of a larger group. The origins of the word are actually from the French word for example, essample.



Software The word software is used today to describe computer programs. It was first used in that sense by Mathematician John Tukey in a 1958 paper for the American Mathematical Monthly (January, 1958). The paper was actually about making applied mathematics more interesting to students, and included both the terms software and hardware. Near the opening he writes

..the "software" comprising the carefully planned interpretive routines, compilers, and other aspects of automotive programming are at least as important to the modern electronic calculator as its "hardware" of tubes, transistors, wires, tapes and the like.
Tukey was well known for his work in statistics in the middle of the 20th century. He died in July of 2000 at the age of 85. He was also the creator of the term "bit" for binary digit. He is also remembered for his introduction of a fast Fourier transform algorithm in association with J W Cooley in 1965.

The word software was also used almost a century earlier in a very different way. Around 1850 software was used to describe garbage which would decompose in a land fill, and hardware was used for items which would not.



Tesseract A tesseract is a name sometimes used for the extension of a cube to four-dimensions. It is also often called a hypercube. Tesseracts were popularized for young readers by the story A Wrinkle in Time in which author Madaline L'Engle uses the idea of a tesseract as a space-time wrinkle that allows cross dimensional travel. If you have trouble visualizing hypercubes, and who doesn't, maybe a look at this page from CUT-THE-KNOT with a computer animation where a single point is extended, dimension by dimension, to become a tesseract will help. The page also has several other good illustrations and links to additional material.

You can explore the impact of multiple dimensions by playing tic-tac-toe on a torus or Kleinbottle at a site created by MacArthur Fellow, Jeff Weeks.

While typing this article I realized that all nine letters of tesseract are typed with the left hand. Can you think of longer words that are typed on a sinlgle hand. The word tesseract comes from the union of the Greek tessares for four and atkis for ray, describing appropriately a solid generated by extensions of rays in four perpendicular directions. It seems that it was coined and first used by Charles Hinton in 1888 in A New Era of Thought.



Viviani's Theorem is named for Vincenzo Viviani(1622-1703), a pupil of Galileo and Torricelli. . The theorem states that in an equilateral triangle, the sum of the perpendicular distances to the sides is equal to the altitude of the triangle. In the figure h=PE+PF+PG. If the point is outside the triangle, the relationship will still hold if one or more of the perpendiculars is treated as a negative value. The theorem can be generalized to a regular n-gon to state, for any point P interior to a regular n-gon, the sum of the perpendicular distances to the n sides is n times the apothem of the figure.

Viviani is also remembered for a reconstruction of a book on the conic sections of Apollonius. He also found a way to trisect an angle using an equilateral hyperbola.