Ed Pegg Jr., May 1, 2007
Mathematica 6 has just been released. It's a great program, but I happen to be an employee of Wolfram Research. I'm reminded of when I saw Penn, of Penn and Teller, juggling three large freshly broken bottles. One of the bottles had a truly ghastly curved spiraling jagged shard. "That's a very nasty looking bottle there!" exclaimed Penn, whirling the particular bottle in front of his face. "That's just part of my normal patter for this act. I always say how nasty the bottles look. But this time, I'm really serious." I believed him. So now, as a biased employee, I'm engaged in predictable patter, trying to lure you into going to the Mathematica 6 launch page. It has an amazing flash banner. And this time, I'm really serious. A small part of the Mathematica 6 launch is the Wolfram Demonstrations Project.
Figure 1: Twelve of the 1200+ available math demonstrations.
The Wolfram Demonstrations Project demonstrates over a thousand mathematical concepts, with more being added every day. Most amazingly, these interactive demonstrations are available free, and can be used by anyone. All you need is the Mathematica Player (also free). There is a wide selection of mathematics on display, and anyone may contribute more.
For example, a calculation can find the scrap metal price of coins. The US Jefferson Nickel is particularly interesting right now -- $1.00 worth of normal nickels has a scrap metal price of $1.88 (as of 1 May, 2007). That's mainly because the metal nickel is near an all time high. With this demonstration, the sliders can be moved to give the current spot price of each metal, and the price of the coins is either the scrap metal price, or $1, whichever is greater. As a side note, it's now illegal to melt US Coins.
Figure 2: The Scrap Metal price of Coins.
Tobias Kreisel and Sascha Kurz found an Integer Distance Heptagon -- a set of seven points, no three on a line and no four on a circle, so that all points are at integer distances from each other. I wanted to make a nice picture of this, and a bit of interactivity helped for instantly labeling all the lines. The existence of an integer heptagon was been a long-unsolved question. Is there an integer octagon?
Figure 3. Labeling the Integer Heptagon.
If you want a statue to look as big as possible, where should you stand? If you are too close, the statue will be foreshortened. If far away, the statue will appear small. In the fourteenth century, Johannes Regiomontanus solved this question. It's a nice geometry problem.
Figure 4. The Statue of Regiomontanus.
Many educational demonstrations are available. For example, the slope-intersect formulas for lines, shown below. This demo is by Abby Brown, of abbymath.com. Abby made many educational demonstrations for her classes, such as conic section curves, projectile motion, illustrating cosine with the unit circle, and solids of revolution. Other people made demonstrations for the chain rule, Riemann sums, arc length, combining functions, Taylor series, and more.
Figure 5. Lines: Slope-Intercept, by Abby Brown.
Various interactive tables of mathematical data are included. For example, the ever popular table of prime factorizations. Demonstrations are available for factoring binomials, how Pratt certificates of primality work, LU decomposition, singular value decomposition, polynomial long multiplication, polynomial long division, and more.
Figure 6. Prime Factorization Table.
An interactive slide rule is available, by George Beck (calc101.com). The slide rule really works, with moving bars. If you need some brush-up on how slide rules work, you can visit multiplication by adding logs, or logplots and beyond.
Figure 7. Slide Rule, by George Beck.
Many interactive puzzles are available. For example, each of the dots within the Orchard Planting Problem can be moved around on the grid, and the generated lines will be calculated automatically. Finding configurations that maximize or minimize lines make very tricky problems (no three in a line). Other problems, such as Planarity and the Bouniakowsky Conjecture, can be tackled.
Figure 8. The Orchard Planting Problem.
There are also many stunning pictures available in mathematics. For example, the relationship between Farey fractions and Ford circles, Cayley Graphs, circles and ellipses, or the Chow-Ruskey order-5 Venn diagram. Sándor Kabai put together many beautiful, interactive demonstrations, such as compound of 5 cubes, twelve decagons, Hoberman sphere, and 120 rhombic triacontahedra.
Figure 9. Buckyball of Buckyballs, by Sándor Kabai
I've given pictures for less than 1 percent of the available material at the Wolfram Demonstrations Project. I look forward to hearing your thoughts about it.
Abby Brown, "abbymath.com," http://teachers.sduhsd.k12.ca.us/abrown/index2.html.
Benjamin Chaffin, "No Three In Line Problem," http://wso.williams.edu/%7Ebchaffin/no_three_in_line/index.htm.
Theodore Gray, "Nickel," http://www.theodoregray.com/PeriodicTable/Elements/028/index.html.
Barbara Hagenbaugh, USA TODAY, "New rules outlaw melting pennies, nickels for profit," Dec 14, 2006, http://www.usatoday.com/money/2006-12-14-melting-ban-usat_x.htm.
Tobias Kreisel and Sascha Kurz, "There are Integral Heptagons," Nov 7, 2006, http://www.wm.uni-bayreuth.de/fileadmin/Sascha/Publikationen2/rare.pdf.
The Wolfram Demonstrations Project, http://demonstrations.wolfram.com/.
Comments are welcome. Please send comments to Ed Pegg Jr. at firstname.lastname@example.org.
Ed Pegg Jr. is the webmaster of mathpuzzle.com. He works at Wolfram Research, Inc. as an associate editor of MathWorld. He is also a math consultant for the TV show Numb3rs.