The Journal of Singularities - Volume 10: The 12th International Workshop on Real and Complex Singularities

The Journal of Singularities

Volume 10

The 12 International Workshop on Real and Complex Singularities

July 22-27, 2012

Celebrating the 60th birthday of Prof. Shyuichi Izumiya, ICMC-USP, São Carlos, Brazil

GO TO JOURNAL OF SING

go to www.journalofsing.org to read this and all preceding volumes

Throwback Fact of the Week - Sir Isaac Newton - 12/25/14

Merry NEWTON-mas everyone! Sir Isaac Newton was born on Christmas Day, 1642, in Lincolnshire, England. Today would be his 372nd birthday!

Newton was one of the most influential and prolific scientists and mathematicians of all time. Among his numerous accomplishments were: laying the foundations of classical mechanics, which include his laws of motion; developing calculus (shared credit with Gotfried Leibniz); and various contributions to the field of optics, primarily concerning the refraction of light.

You may note that we left out (arguably) his most famous work: the formulation of the Law of Gravity. As William Stukeley, one of Newton's first biographers, put it:

"He was in the same situation, under the shade of some apple trees, the notion of gravitation came into his mind. It was occasioned by the fall of an apple, as he sat in a contemplative mood. Why should the apple always descend perpendicularly to the ground, thought he to himself. Why should it not go sideways or upwards, but constantly to the earth's center? Assuredly, the reason is that the earth draws it."

Sir Isaac Newton

While Newton's place in history is monumental, let's not forget one of his most famous and humble quotes, "If I have seen further it is only by standing on the shoulders of giants." The slight irony is that while this is true, he himself became one of those giants on which later generations would stand and push further the boundaries of science. 

Throwback Fact of the Week - Fractals - 12/18/14

Simply put, fractals are a never-ending pattern. Fractals are often complex patterns that are self-similar (look the same) at any scale. Essentially, when you zoom in or out on a fractal image, you expect to see the same pattern or at the least a very similar one. 

This may not sound like a precise definition; that's because it's not. A formal definition of fractals has not been settled upon. However, the man who coined the term fractal, Benoît Mandelbrot, once described them as, "beautiful, damn hard, increasingly useful. That's fractals."

The math behind these incredibly beautiful objects is outside the scope of this blog post (particularly fractals involving the complex plane). So we will end by simply marveling at a couple of these fractals, to help get a better conceptual idea. 

Koch Snowflake: formed by taking a equilateral triangle and replacing the middle third of each line segment with a pair of equal line segments that form the next "triangle"

Image credit: Maksim. From left to right: Mandelbrot set normal zoom, same set at x6 zoom, and same set at x100 zoom. 

Throwback Fact of the Week - Pick's Theorem - 12/11/14

Pick's Theorem is a clever way to calculate the area of a simple polygon. All you need is a pencil and some graphing paper to see it come to life and simplify some potentially monotonous calculations.

The method is as follows:

1. Begin by drawing a simple polygon on an equally spaced grid (i.e. graphing paper) so that all its vertices lie on grid points.
2. Count the number of grid points, i, located in the interior of the polygon. Then, count the number of grid points, b, that fall on the boundary of the polygon. 

Pick's Theorem tells us that the area (in units squared), A, of the polygon can be calculated neatly as: 

A = i + b/2 - 1.

Note that this Theorem applies only to simple polygons, those with no holes and consisting only of one piece. 

i = 7, b = 8
A = 7+8/2-1 = 10

A useful and handy application of this theorem is roughly estimating an area on a map (of say a region or country), by overlaying a grid and using a polygon to approximate the shape of the region you are interested in. 

The Holiday Math Gift Guide: Best Gifts for the Math Enthusiast in Your Life!

 The Holiday Math Gift Guide: 
Best Gifts for the Math Enthusiast in Your Life!

     Happy Holidays! Are you looking for gift ideas for the mathematician in your life this holiday season? Are you a mathematician yourself, making your wish-list and checking it twice? Then this holiday gift guide for the mathematician can help!

     A simple search online will return hundreds, if not, thousands of math-related clothes. Shirts and hats are most-popular and commonly feature mostly-tired math jokes and puns. If you're recipient is a new or young enthusiast, you might add this classic ($23.95, Zazzle) or one like it to their wardrobe. Here's an everyday hat featuring Schrödinger's wave equation ($21.95 at Zazzle).

     When considering clothing, we encourage you to step it up. Accordingly, this year we most like the Mathematics Ugly Christmas Sweater ($34.95, TeeSpring). This 100% cotton, USA made beauty is sure to turn your mathematician into the talk of the party, class or study group. Dealing with somebody who's wishing for a more hands-on present? Why not set them out to knit their own ($16.41, Amazon) ugly math sweater. May we recommend Σ (notably missing from TeeSpring's offering) for your design?

Ugly Math Sweater

     Actually fashionable, Doug McKenna's scarves ($77.00, DMCK Designs) feature fractal tiling patterns inspired by the space-filling curves introduced by 19th century Italian mathematician, Giuseppe Peano.

     For more math clothing ideas, check out Algebra Cat ($22.40, Redbubble), the Center of Math ($9.95, Center of Math) and for the mathematician who's expecting, this cute onesie from TreeHouse Apparel ($14.00, Etsy).

Algebra Cat

     Mathematicians are problem solvers. They like to be challenged and stay sharp. The games and toys you give them should help exercise their minds. We've chosen just a few of our favorites to list here.

     First, mathematicians need peace and quiet in order to exercise the mind. True to ancient Roman design, the Exclusive Wooden Catapult Kit ($29.99, ThinkGeek) is a must-have toy/tool for the mathematician's work space. Loud neighbor? Somebody stealing all your pencils? Once constructed, this catapult will lob objects over 20 feet and is perfect for protecting pencils and peace. If building a toy sounds good, check to see if this Wooden Mechanical Clock Kit (sold out, ThinkGeek) is back in stock.

Catapult

     We had fun watching Maths Gear explain their Shapes of Constant Width ($12.29, Maths Gear).

     Now, games, particularly strategy games, are tremendously popular. If you're unfamiliar with 2048, Sudoku, STRATEGY, you should start there. We love Perplexus, the makers of brain teasing maze and sequential puzzles. Check out the Perplexus Epic ($28.80, Amazon) for any patient, determined mathematicians you may know. The company advertises the toy as being "like a puzzle inside an enigma, wrapped in a maze, on a date with a riddle...at a confusion convention." Cool. Traditional games too easy or old? Why not mix it up with a pair of Sicherman Dice ($6.25, Grand Illusions)? You can use these unique dice to play most of the games you normally would without changing the odds!

     First, here's some of this year's most-desired electronic video game gifts, liked by mathematicians: Civilization V, Minecraft, and games from Blizzard Entertainment (including: World of Warcraft, Starcraft 2 and Diablo 3.)

     Of course, to play them you'll need a gaming console or PC. Industry leaders Sony and Microsoft are dueling this season, selling Playstation 4's ($419.99, Amazon) and Xbox One's (349.99, Amazon) respectively. Both are good options but do your homework before buying! For mathematicians interested in gaming and code you should consider buying a PC instead or kit to build your own! The Dell Alienware Area 51 (2014) ($1,699, Dell) is an absolute beast of a machine. New Egg is a good place to buy the parts you'll need to build your own machine. If that's not enough to keep your gamer busy, also gift them Matrices, Vectors and 3D Math: A Game Programming Approach With MATLAB ($9.95, Center of Math).

     For the mathematician excited by aeronautics, we recommend the entry-level Parrot Minidrone Rolling Spider ($99.99, Parrot).

Parrot Minidrone

     Give the gift of a WolframAlpha Pro subscription ($5.49/mo., WolframAlpha). It's perfect for students and is sure to make any mathematician smile.

     None of the above electronics options above have your hair sticking straight up? You could give the gift of a math-movie pack! Check here for Wikipedia's list of films about math and here for Harvard University's Prof. Oliver Knill's "Mathematics in Movies" page. If none of these things make your mathematician happy, simply show them this video explaining the algorithms behind "hitting it off." Offer to pay the registration fee, if there is one. Math and match-making. What's not to love?

     Every mathematician needs a neat (read: cool, not necessarily clean) workspace. Take a look at our best guesses to fill the office or workspace:

     Start with a gift certificate to the preferred caffeine supplier in your local area. For Center of Math employees in New England, that means you're choosing either Dunkin' Donuts or Starbucks gift cards. Each company offers unique features and rewards if the card-holder links the preloaded gift card to the company rewards program. Many mathematicians require coffee. Coffee requires a good mug. We recommend the Chalkboard Mug ($14, Exploratorium) which will allow for uninterrupted sipping and note-taking. If scribbling on a liquid-filled mug sounds like asking for trouble to you, simply go with this classic.

Engineer Mug

     If you have the space, we really like the Offex Mobile Double-sided Magnetic Whiteboard ($299.99, Amazon). Whiteboard or blackboard? You had better check before buying -- some mathematicians won't work on one or the other! We also like Magic Whiteboard ($66.99, MagicWhiteboard) for more confined spaces or the traveling mathematician. Like we recently did in the Center of Math's Studio Classroom, you could upgrade the furniture your mathematician works in! Here's the comfy LexMod mesh chairs our visitors now sit in.

     Thinking smaller (and cheaper): Check out the pure math sculpture art at Bathsheba Sculpture or Henry Segerman's Mathematical Art shop. Or, how about a gift to make your own hypotrochoid art ($8.00, Uncommon Goods). Now you've got the desk and walls covered. How about some plant life? We'd argue the best buy for a mathematician is Sprout Growing Pencils ($24.99, ThinkGeek).

     Finally, a subscription for ad-free music streaming will keep your mathematician moving and grooving. Try Spotify, Pandora or Rdio.

     Any mathematician would appreciate the opportunity to join the American Mathematical Society ($69/yr., AMS) or Mathematics Association of America ($169/mo., MAA) free of charge. Like the Dunks' vs 'Bucks (not sure anybody calls it that) debate above, each organization is unique and you should explore both before purchasing your membership.

     If you had something a bit more grand in mind, consider sending your mathematician on a math-inspired trip! Not sure where to go? San Antonio, TX will host the Joint Mathematics Meetings 2015 in January. Find out more here. MAA's MAAthFest will be in Washington, D.C. in August, 2015. The Center of Math will be at both!

     Texas too big or too far? Consider sending your math-maniac to the one and only Cambridge, MA. The Boston-Cambridge Metro area is ideal for the traveling math enthusiast. Like baseball (the statisticians game)? Triple Crown Travel's East Coast Swing ($2,495, Triple Crown Travel) would bring you here. Should you decide to make the trip, be sure to come find us!

     Let's assume you're unamused. Here's our gathering of nearly-random, miscellaneous math gifts to consider:

     Time-telling: Ever look at a mathematician's clock? Get one here. How about a watch? Try the Maths Equation Watch for Mathematics Nerds ($49.95, Zazzle) or the Prime Time Watch ($38, Uncommon Goods).

     Pi Bowls: Pi Day is rapidly approaching! We like the idea of serving company snacks in the Pi Bowl or Pi Basket. Speaking of Pi, here's an inspired shower curtain.

Pi Shower Curtain

     Lights: 'Tis the Season for lights! We think you should ditch your bulb and try Plumen 001 designer bulbs ($34.95, Plumen). We also came across Studio Cheha Flat LED lamps ($120, Studio Cheha) and had to share.

And last but not least!

     One of the top gifts for mathematicians this year, of course, will be books! We left books off our list. The only books we'll mention here are the always affordable, always accessible digital and print textbooks published right here at the Center of Math ($9.95/29.95, centerofmath.org/store). Oh, and this one: Math for the Frightened: Facing Scary Symbols and Everything Else That Freaks You Out About Mathematics.

Did we miss something? Think the mathematician you know would hate this list? Help us get it right in the comments!

Happy Holidays from your very merry math friends at the Center of Math!

Throwback Fact of the Week - Ishango Bone - 12/04/14

The Ishango Bone was discovered by Jean de Heinzelin de Braucourt, a Belgian geologist, in 1960.

The bone (pictured below), estimated to be more than 20,000 years old, is a baboon's fibula and was discovered to have many markings engraved into it. The bone was first thought to be a simple tally stick. However, upon further investigation, many scientists and mathematicians believe that the markings are indicative of a mathematical understanding that transcends basic counting.

Some of the more striking features of the bone include a column of three notches that double to six, a column of four notches that double to eight, and ten notches that halve to five; these are all indicative of a basic understanding of doubling or multiplication. 

Even more curious is the fact that all the numbers in the other columns are odd and one of those columns consists of all the prime numbers between 10 and 20. The fact that there are prime numbers clearly separated would indicate some understanding of division. 

There are tally sticks that have been discovered that predate the Ishango Bone. However, the Ishango Bone is the oldest one known that contains logical carvings giving evidence of a deeper mathematical understanding.

Throwback Fact of the Week - Magic Squares - 11/20/14

A magic square consists of a square grid with n rows and n columns, thus it consists of  n² total boxes. These boxes are all filled with different integers. The sums of the numbers in each of the vertical columns, horizontal rows, and the diagonals are all equal.

A magic square is said to be normal if the integers filling the boxes are the consecutive integers from 1 to n². The sum, S, of the rows (columns and diagonals) for a normal magic square is given by the formula: S = n(n²+1)/2

The normal 4x4 magic square below was created in 1514 by Renaissance artist Albrecht Dürer in his engraving titled Melencolia I. 

This square has many amazing and intriguing properties. The sums of each of the rows, columns, and diagonals is 34 (as expected from a normal 4x4 magic square). However, notice that the bottom two central numbers are 15 and 14, i.e., the year the engraving was made, 1514. The numbers in the bottom left and right corners, 4 and 1, correspond alpha-numerically to the engraver's initials A.D.! 

Additionally, the sum of the four corners (16+13+1+4), the sum of the middle 2x2 square (10+11+6+7), the sum of the middle two entries of the two outside columns and rows (5+9+8+12) (15+14+3+2), are all equal to 34. 

Throwback Fact of the Week - Knight's Tour - 11/06/14

Knight's Tours have fascinated mathematicians for centuries. A Knight's Tour is a set of moves made by the knight on a chessboard, on which the piece visits each square exactly one time. Leonhard Euler was the first to write a mathematical paper analyzing these tours. 

If you are unfamiliar with chess, the knight piece may move in an "L" shape; it can move two squares horizontally and one square vertically, or two squares vertically and one square horizontally. 

A tour is called closed or reentrant if the knight ends its tour on a square that is one move away from the square it started (meaning it could begin the same tour over again). Otherwise it is an open tour. 

Apart from the standard 8x8 square chessboard, Knight's Tours have been studied on boards with varying dimensions as well as on irregular (not square) surfaces. 

Below is an animation of an open Knight's Tour on a 5x5 chessboard. 

The Worldwide Lecture Seminar Series Presents: Alexandru Dimca

CAMBRIDGE -- Local and traveling mathematicians gathered Friday at the Center of Math where the Worldwide Lecture Seminar Series presented Alexandru Dimca, Université Nice Sophia Antipolis, on the fundamental groups of complex algebraic varieties. The event was captured in its entirety and has been made available free to the public.

For more Worldwide Lecture Seminar Series coverage please click here.

The Worldwide Lecture Seminar Series Presents Alexandru Dimca on the fundamental groups of complex algebraic varieties.

Una foto pubblicata da Worldwide Mathematics (@centerofmath) on Nov 11, 2014 at 8:33 PST

Throwback Fact of the Week - Sicherman Dice - 11/06/14

Sicherman dice are not your typical board game dice, but you could use them in almost any board game without changing the outcomes. How can this be? 

The faces on the Sicherman dice (pictured below) are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8. Sicherman dice are very unique; they are the only pair of 6-sided cubic dice with positive integers that have the same odds as your standard cubic dice. Note, if negative numbers are allowed there are an infinite number of such dice.

Image credit: Grand Illusions (dice can also be purchased from their site)

What does this mean? As you know, with regular dice there is a fixed probability of rolling a given sum (you can roll between 2 and 12) when both dice are thrown. The same holds true for Sicherman dice; you can roll a sum between 2 and 12 and the odds of a particular sum occuring is the same as for a regular pair of dice. 

For example, regular dice have a 1/9 chance of rolling a sum of 5. With Sicherman dice, there is also a 1/9 chance of rolling a sum of 5. 

We mention you could play almost any game with these. The reason for almost is because some games have certain rules when doubles are rolled and these dice have different odds of doubles being rolled. The reader is left with the exercise of calculating the odds of rolling doubles using Sicherman dice!

Throwback Fact of the Week - Vampire Numbers - 10/30/14

Vampire numbers were first defined by Clifford Pickover in 1994. 

A vampire number is a composite whole number with an even number of digits, n. The vampire number has form v = xy where the digits x and y each have n/2 digits. These digits consist only of the digits in the vampire number, but in any order. Note that x and y are not allowed to both end in trailing zeros. 

1260 is the smallest vampire number.

 v = xy

1260 = 21*60

You'll notice that in this case, the vampire number has 4 digits; x and y each have 2 digits, and those digits consist of a re-arrangement of the digits in the vampire number. 

The digits x and y are known as the fangs of the vampire number. Vampire numbers sometimes have multiple pairs of fangs that fit the rules for forming the vampire number. Such as 125460 = 204 × 615 but it also equals 246 × 510.

An even more interesting type of vampire number would be a vampire number where its fangs are also its prime factors. For example, 117067 = 167 × 701, where 167 and 701 are the prime factors of 117067, making this number a prime vampire number! 

Throwback Fact of the Week - Benford's Law - 10/23/14

Benford's Law, or more descriptively, the first-digit law, states that in most number lists and real-life collections of data, (such as death rates, baseball statistics, etc.) 1 occurs in the leftmost leading position roughly 30% of the time. It also asserts that larger digits occur less and less frequently, with 9 occurring a mere 4.6% of the time as the first digit.

The significance of this law lies in the fact that it shows that the most common first digit in an arbitrary source of data is not random; if it were random, every digit would be expected to occur first about 11.1% of the time, a 1 out of 9 probability.

The graph below shows probability that each digit occurs as the leading digit. A given set of numbers satisfies Benford's Law if the leading digit d, where d can be the digits 1 through 9, occurs with probability log₁₀(1+1/d).

The law is named after physicist, Dr. Frank Benford, who stated it in 1938. However, it had been previously pointed out by Simon Newcomb in 1881, when he noticed that the pages of logarithms containing the numbers beginning with 1 were much more worn out than other pages.

The law has been used as a method to detect fraud; for example, an accountant could detect a fraudulent tax document if the occurrence of leading digits does not sync up closely with Benford's Law.

Throwback Fact of the Week - Zero - 10/16/14

Zero. We often overlook the importance of this number and its existence does not seem foreign or bizzare to us... anymore.

The Greeks lacked a symbol and concept for zero. They wrestled with the philosophical implications of how nothing could be something.

The Babylonians developed an excellent sexagesimal number system (base 60) but they lacked the concept of zero. Over time, the Babylonians would develop a system of using a space as a placeholder between digits that functioned similarly to the modern zero. However, this space was not zero; the concept of representing nothing was foreign to them. Without the idea of zero, there was no way to distinguish certain Babylonian numbers, just like how today we wouldn't be able to distinguish the numbers 13, 103, 130 10003 without the use of zero. 

The development of zero as an actual number (not just an empty space) is credited to 7th century Indian mathematics. Around the same time, the concept was being used by the Mayan civilization. However, it was the Indian concept of zero that spread to Arabia, Europe and China. 

Indian mathematics treated zero like any other number, using it in calculations (even in division). It was the Indian mathematician, Bragmagupta, who first laid out a set of rules governing the use of zero. These rules included things like a number subtracted from itself is zero and a number times zero is zero. 

The importance of zero can be summed up by Pierre-Simon Laplace, who said:

"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated ... The importance of this invention is more readily appreciated when one considers that it was beyod the two greatest men of antiquity, Archimedes and Apollonius."

Center of Math NEWs: Greg Moore - Physical Mathematics and the Future

Center of Math NEWs brings you fun and interesting new mathematics news from around the world.

Topics: Physical mathematics, string theory, infinite-dimensional Lie algebras, geometry, analytic number theory, topology,  quantum gravity, string theory, supersymmetry

Physical Mathematics and the Future - Gregory W. Moore: paper (PDF)

Show slides: here 

 Runtime 16:02

Strings 2014: go

Throwback Fact of the Week - Viviani's Theorem - 10/09/14

Viviani's Theorem can be broken into 2 steps:

1. Choose a point anywhere inside of an equilateral triangle
2. Draw perpendicular lines from the point to each of the 3 sides of the triangle

The theorem states that the sum of the lengths of these lines is equal to the height of the triangle. Using the image below, the theorem states that x + y + z = h, no matter where inside the triangle you place point P. 

The theorem was proven by Vincenzo Viviani around the year 1659. The proof can be derived easily from the formula for the area of a triangle (Area = .5bh, where b is the base of the triangle and h is the height). To quote the typical math textbook, "this proof is left as an exercise for the reader."

This theorem can be generalized to any regular n-sided polygon. For the case of an n-sided polygon, the sum of the perpendicular distances from an interior point to each of the n sides is equal to n times the length of the apothem of the polygon (Recall that an apothem of a regular polygon is a perpendicular line segment from the center of the polygon to the midpoint of one of its sides).

Throwback Fact of the Week - Tower of Hanoi - 10/02/14

The Tower of Hanoi is an intriguing puzzle game made popular by French mathematician, Edouard Lucas, in 1883. 

The puzzle consists of disks of different sizes that can be moved onto any of the 3 pegs. At the start, the disks are all stacked on the leftmost peg in order of size, with the smallest at the top. The goal of the puzzle is to move the entire stack to the rightmost peg, while obeying these 3 simple rules:

1. Only one disk at a time may be moved to another peg 
2. You can only move the top disk from a stack of disks
3. A larger disk cannot be placed on top of a smaller disk

The minimum number of moves necessary to solve the puzzle with n disks is 2ⁿ - 1. There are simple algorithms and strategies for solving this puzzle in the minimal number of steps. 

Check out the video below to see the game being played optimally starting with 5 disks (2^5-1 = 31 moves). 

Video Credit: Mohammad Al-Khanfar

New Center of Math Animation!

     These past few weeks have been busy at the center; with our followers rapidly growing, we decided to make a short video to thank everyone for their support! After days of toiling with cameras, lighting, and crazy setups, we finally finished our own whiteboard animation. Come check out our Youtube channel to see our new video!

Previously mentioned ridiculous setup...

Throwback Fact of the Week - Collatz Conjecture - 9/25/14

Collatz Conjecture

This conjecture is named after German mathematician, Lothar Collatz, who first proposed the problem in 1937. 

Start with any positive integer   , then apply the following recursion:

Credit: mathworld.wolfram.com

The Collatz conjecture states the following: regardless of which positive integer is chosen initially, this sequence of numbers will always eventually reach 1.

For example, starting with 10 (which is even, so divide by 2) you get the following sequence of numbers (5,16,8,4,2,1). Or starting with 7 you get (22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1). Note that upon reaching 1 you then get (4,2,1,4,2,1...) indefinitely.  

Despite the apparent simplicity with which the problem is stated, there has been no proof that every positive integer will eventually reach 1 using this recursion. While there has been no proof that it works for every integer, it has been checked to work for every number up to 5.764×10¹⁸

The Collatz Conjecture has fascinated mathematicians for decades. Mathematician Paul Erdos even commented on the complexity of the problem by saying "Mathematics is not yet ready for such problems."

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Throwback Fact of the Week - HP-35: The First Scientific Pocket Calculator - 9/18/14

HP-35: The First Scientific Pocket Calculator

In 1972, the Hewlett-Packard company (commonly known as HP) introduced the first ever scientific pocket calculator, called the HP-35. The term scientific meant that it was able to calculate trigonometric and exponential functions. The HP-35 was the first ever calculator to do these calculations; prior calculators only performed addition, subtraction, multiplication and division.

Before the introduction of the HP-35, trigonometric and exponential calculations were done using a slide rule, which were only accurate to a few significant figures. The fact that you may have never even heard of or seen a slide rule before is because the introduction of this calculator made slide rules obsolete.

An HP-35 Scientific Pocket Calculator
Image Credit: Seth Morabito 

The name HP-35 came from the 35 keys on the device. It was only anticipated to sell about 10,000 units in the first year. However, HP greatly underestimated the demand for such a device and ended up selling over 100,000 in the first year.

Throwback Fact of the Week - Euler's Identity - 9/11/14

Euler's Identity

Euler's Identity is considered to be one of the most amazing and beautiful mathematical relationships ever discovered. The statement of the equality is as follows:

You may wonder why this equality is looked upon with such awe and admiration.

The identity seamlessly links some of the most important mathematical symbols: the number 0 (the additive identity), the number 1 (the multiplicative identity), the transcendental numbers π and e (Euler's number), and i (the imaginary unit). If this wasn't enough, it also makes use of three of the basic arithmetic operations: addition, multiplication and exponentiation.

The derivation of the identity follows from Euler's formula, which states:

Evaluating this formula at x = π yields the identity. 

Kasner and Newman note, "We can only reproduce the equation and not stop to inquire into its implications. It appeals equally to the mystic, the scientist, and the mathematician."

Throwback Fact of the Week - Wheat on a Chessboard - 9/4/14

Wheat on a Chessboard

This 800-year-old problem in mathematics is notable because of the fluency at which it illustrates the startling nature of geometric growth.

The legend goes that the Indian King, Shirham, asked his Grand Vizier, Sissa ben Dahir, what reward he would like for inventing the game of chess. The Grand Vizier asked for one grain of wheat for the first square of the chessboard, two grains for the second square, four grains for the third, eight grains for the fourth square, and so on for the sixty-four total squares. This appeared to the king to be a small and foolish request, so he granted the grand vizier's wish.

Little did the king know just how many grains of wheat would accumulate by the 64th square. This number would be the sum 1+2+2²+2³+…+2⁶³ = 2⁶⁴-1. This comes out to be 18,446,744,073,709,551,615 grains of wheat (over 18 quintillion!). Clearly the king would run out of grain long before this total. This is enough wheat to cover the entire earth several inches deep!


Watch the video below to get an illustration of this legend and the absurd amount of wheat!

Video Credit: Jill Britton

Throwback Fact of the Week - Sieve of Eratosthenes - 8/28/14

Sieve of Eratosthenes

Quickly recall that a prime number is a number larger than 1 that is divisible only by itself and 1 (e.g. 2, 5, 7, 11...). Euclid showed that there are an infinite number of primes. However, it wasn't until around 240 B.C. that Eratosthenes developed the first known method for finding primes, a method known today as the Sieve of Eratosthenes.

The Sieve can be used to find all prime numbers up to a given integer, n. 

The method is straightforward: make a a list of all the integers less than or equal to n, cross out the multiples of all primes that are less than or equal to the square root of n, the numbers that remain are the primes. Typically it is easiest to start with 2, cross out all multiples of 2. Then 3, cross out all its multiples etc. 

An example with n = 120 is demonstrated the video below. The video crosses out the multiples of 2, then of 3, then of 5 and lastly the multiples of 7 (since 7 is the largest prime that is less than the square root of 120). After 7, the video highlights the remaining primes in pink.

By Ricordisamoa (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Prime numbers continue to fascinate mathematicians today. Some of the most famous unsolved problems in mathematics center around prime numbers, including the Goldbach Conjecture and the Riemann Hypothesis. 

Throwback Fact of the Week - Perfect Numbers - 8/21/14

Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its proper positive divisors. In other words, it is equal to the sum of its positive divisors excluding the number itself. The simplest example of this is the number 6; the proper divisors of 6 are 1, 2, and 3, 1+2+3=6, so 6 is a perfect number. 

Knowing that 6, which is a small integer, is a perfect number might lead one to believe that numbers of this form are common. This turns out to be false; they are much rarer. The next perfect numbers are 28, 496, and 8128. The next perfect number after 8128? 33,550,336. 

Perfect numbers are not new. Pythagoras and the Greeks knew about the first four perfect numbers (you can see how they might not have encountered or been able to find the factors of a number like 33,550,336). 

There have been a number of interesting results about perfect numbers. The most important result was initially discovered by Euclid and later expanded upon by Euler.

Euclid proved that if if 2^p-1 is prime, then N = (2^p-1)(2^p-1) is an even perfect number. 

Prime numbers of the form 2^p-1, where p is prime, are known as Mersenne primes, another bountiful world of study themselves. Two millennia later, Euler proved that (2^p-1)(2^p-1), where 2^p-1 is a Mersenne prime, will produce all even perfect numbers (there is a 1-1 ratio).

It remains unknown if there are an infinite number of Mersenne primes (only 43 Mersenne primes are known so far). It also remains unknown if there are any odd perfect numbers.

#textbooksareexpensive

Don't let this happen to you. DO the math. DON'T overpay.

Throwback Fact of the Week - This Day in History: Guido Castelnuovo - 8/14/14

On this day in history

On August 14th, 1865 Guido Castelnuovo was born in Venice, Italy. He was an Italian mathematician who made contributions to the fields of algebraic geometry and statistics. 

Despite being forced into hiding during World War II, Castelnuovo organized and taught courses to Jewish students who were forbidden from attending university. 

After the war, he was charged with repairing the damage done to Italian institutions throughout the twenty years of Mussolini's rule.

Throwback Fact of the Week - Golden Ratio - 8/07/14

The golden ratio is a deceptively intriguing bit of mathematics. The ratio makes numerous appearances throughout history in architecture, nature, art, music, and most importantly mathematics. 

It is easiest to understand the ratio by imagining a line. Divide the line into two segments with long segment a and shorter segment b. The segments are said to be in the golden ratio if the ratio of the whole line to the longer part is the same as the ratio of the longer part to the shorter part, or simply if (a+b)/a = a/b. This ratio comes out to be an irrational number (approx. 1.61803...), known as the golden ratio, and is symbolized by the Greek letter φ (phi). 

If the lengths of the sides of a rectangle are in the golden ratio, the rectangle is called a golden rectangle. This rectangle exhibits some interesting properties. It is possible to divide a golden rectangle into a square and another (smaller) golden rectangle (pictured below). This process can be repeated indefinitely.

A logarithmic spiral can be drawn and this spiral will closely approximate the Golden Spiral, which is a spiral that gets wider by a factor of φ with ever quarter turn.

Throwback Fact of the Week - Newcomb's Paradox - 7/31/14

Newcomb's Paradox

Newcomb's Paradox is a thought experiment devised by William Newcomb in 1960.

The problem has taken on various forms over the years but the general idea(s) remain in place.

Before you are two boxes: one transparent (box 1) that always contains $1,000 and one opaque (box 2) which either contains $1,000,000 or $0. An entity, often called the Predictor, is exceptionally good at predicting people's actions, the Predictor is almost never wrong. The Predictor explains that you have two choices: take what is in both boxes, or take only what is in the opaque box, box 2. 

There is a twist; the Predictor has made a prediction about what you will decide. If the prediction is that you will take both boxes, then $0 will have been placed in the opaque box. If the prediction is that you will take only the opaque box, then $1,000,000 will have been placed inside of it. By the time the game begins, the prediction has already been made and the contents of box 2 already determined. 

Which box do you choose? 

In a 1969 article, philosopher Robert Nozick noted that  "To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly."

To this day there is much disagreement on the best strategy.