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

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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.