Exploring Mathematics

Allen Klinger, © 10/07/2015

This is a collecion of topics in and comments on the field of math. It begins with a large number of links (also called pointers and anchors). All three terms come from Computer Science.

By activating a link one can see an item from many different fields, for example commerce, games, and military activity.

Pointers on this page lead to math items at different levels. Many begin with easily understood material. Some could be used in school mathematics. Their purpose is to stimulate interest in basic concepts like quantity and shape.

This page supports trying new starts in math learning. That people can do math but think they can't, makes many unable to work with technical devices (e.g., computers).

Daily life uses numbers like twenty-four, twelve, sixty and three hundred sixty. Looking at them as in hours in a day (24), or examining the number of eggs in a cardboard carton (12, a dozen), minutes in an hour (60), and degrees in a complete rotation (360), can build numerical capabilities.

People often do things that have math background. They play card games, are active in sports, gamble, make health decisions, and do many other practical things (like buying insurance) that involve mathematics.

Items posted here could help kindle interest in starting math again. That is, learning in a less formal way.

Need for mathematical knowledge happens. That tells us we need learning activities in mathematics.

A rough idea of how to start such an activity begins in a recent paper[15] latest version. Large numbers of topics and the cultures that gave rise to them appear at St. Andrews Math History Topics. The geometric questions that arise in an ancient puzzle are another possible start point.

There are many questions, I've considered: e.g., how to make an accurate approximation to one or another irrational number. For instance, to see how to do that for the base of natural logarithms got to e by powers.

Suppose you want to get started growing math capability. It could help if you were able to tell when some statement is plain wrong. Even when you're not sure which of two others is right, labeling the one that is wrong should get some credit. To do that use two diagrams reachable from clicking the link try.

The diagram on the left is in color. It shows six differently-shaded colored (blue, red and rose), three- (triangular) and four-sided (rectangular) regions.

The middle of the diagram at left has six corners; its shape and name follow

The Cuboctahedron is a very similar solid that was known to Archimedes. Like the image here it has "fourteen bases ... eight triangles". Instead of the Cuboctahedron's "six squares" the solid shown has that number of rectangles.

We could ask questions about these solids. In the image reached by the following link, try, the right diagram is black and white, with two kinds of lines, solid and dashed. The labeled points are each assigned one of the thirteen letters from a to m. The purpose is to allow a person to reflect what he or she believes regarding three simple statements characterizing a situation (like these solids).

If the light blue and light red regions were moved they and the light rose shape could make a six sided planar (lying in a flat plane) shape. Let's call the statement "Left figure has a six sided shape" the a label. (It is wrong, but if movement were allowed it could be right.) Let's call the statement "There are only triangular or three-sided and rectangular or four-sided regions in the colored part of the two diagrams," the b label. If c stands for "All the regions are three-sided in the colored part of the two diagrams," we know that is wrong but some people might not realize that.

Any answer on a line touching b (the correct answer) should get some credit. Any answer on a line between the two incorrect answers a and c should get nothing. Finally an answer of "I don't know," m should be rewarded somewhat (for honesty).

Links tagged with * show untraditional sources. Many are like this dancer: dancer they move (are animated). Some speak. Others jump between topics or are unconventional beginnings. All could be part of offering new entry points to math. They are prototypes for using the world-wide web to enable re-entering the ideas of mathematics.

Italic links go to new items I've written. Other links are to student activities I've led. Others are things I've found on the web that could be worked into new K-12 class activities.



Night Sky*



Math Site





Islamic Mathematics



Ancient, Origins (Babylonian)





Nested Graphic*

Nested Analytic*

Pi and More*[26]








Computer Sources


Ways to Number





Math Sources




Sphere's Insides*[16, 17]













Collatz Problem[2]

Collatz Paper[5]

Make Change[14]

Make Change Solution[14a]

Math Reference

Interactive Math

Sum Integers' Powers[7]



Modern Mathematics




About Nine[13]



[1] Devlin, Keith, The Millennium Problems, NY: Basic Books, 2002, p. ix.

[2] Weisstein, Eric W., World of Mathematics http://mathworld.wolfram.com/, published as CRC Concise Encyclopedia of Mathematics, Second Edition; also see Previato, Emma (Editor), Weisstein, Eric W., Dictionary of Applied Math for Engineers and Scientists (Comprehensive Dictionary of Mathematics); http://mathworld.wolfram.com/CollatzProblem.html

[3] Black, Paul E., Dictionary of Algorithms and Data Structures, http://www.nist.gov/dads/ .

[4] Joyce, David E., (Dept. of Mathematics and Computer Science , Clark University, Worcester, MA), Euclid's Books, Guide, http://aleph0.clarku.edu/~djoyce/java/elements, e.g., http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html

[5] Lagarias, Jeff, "The 3x+1 problem and its generalizations," American Mathematical Monthly, 92, 3-23, 1985.

[6] http://www.gap-system.org/~gap/ and http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html, University of St. Andrews, Scotland.

[7] Weisstein, Eric W., "Faulhaber's Formula," MathWorld--A Wolfram Web Resource, http://mathworld.wolfram.com/FaulhabersFormula.html.

[8] "The On-Line Encyclopedia of Integer Sequences," originated by N. J. A. Sloane at AT&T Research; Welcome to OEIS;

[9] Nelsen, Roger B., Proofs Without Words, The Mathematical Association of America, 1993.

[10] Conway, John H. and Guy, Richard K., The Book of Numbers , Springer-Verlag, 1996.

[11] http://www.cut-the-knot.org/index.shtml, Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny.

[12] http://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art, The Nine Chapters on the Mathematical Art, from China, 1st century AD (perhaps 200 BC).

[13] http://en.wikipedia.org/wiki/Nine

[14] http://www.siam.org/journals/problems/downloadfiles/02-005.pdf; [14a] Jonas Degrave's 14 page solution, http://www.siam.org/journals/problems/downloadfiles/02-005s.pdf

[15] Klinger, A., "Patterns in Numbers," Dresden International Symposium on Technology and its Integration into Mathematics Education 2006, Proceedings of the Conference DES-TIME_2006, ISBN 3-9011769-59-5; also see http://www.cs.ucla.edu/~klinger/patterns_2_2_06.pdf; and later drafts: .../patterns_5_23_07.pdf; .../cm.pdf.

[16] http://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg

[17] http://www.geom.uiuc.edu/docs/outreach/oi/waves/

[18] Klinger, A., "The Vandermonde Matrix," American Mathematical Monthly, 74:5, 571-574, 1967.

[19] Klinger, A., Brown, T., "Problem: Monotonicity of Two Functions," SIAM Review, 10: 452, 1968.

[20] http://www.cdli.ucla.edu/wiki/index.php/Babylonian_mathematics

[21] http://www.stetson.edu/~efriedma/numbers.html

[23] Klinger, A., "Training and Thinking," http://www.cs.ucla.edu/~klinger/training.html, The Tau Beta Pi Bulletin, LXXV, 3, March 2002, pp. 3-5.

[24] http://mathworld.wolfram.com/NicomachussTheorem.html

[25] IEEE Spectrum, April 2007, p.8 "... 727 has the property that its square is the concatenation of two consecutive numbers ... " (727^2 = 528,529) "... interesting facts about some of the numbers from 0 to 9999 ..."

[26] http://www.cs.ucla.edu/~klinger/pipaper.html is the draft paper "Collected Items On Pi - Circle Circumference/Diameter Ratio π and Calculating an Infinite Product - A Background Study Involving Computer Science Undergraduate Students or What We Learned About π and Why We Looked at a Slowly-Converging Infinite Product Π "

[27] Spatial Puzzle due to Hess, Winter 2010, Tau Beta Pi Bent, Brain Ticklers Column


+The voice saying seven key terms in Listen and the next two files, Probability and Multiply, is Jennifer (Jen, Hsu-hua) Chen; the animations in the latter two were by Yu-Chian Tseng. Binary and Area include work by Navid Aghdaie and Dorene Lau whose projects are Binary and Women In Math Careers; the latter has four parts:
Right Triangle Triangles in Space Round - Ellipse Repeat - Recursion
A visual excerpt from Right Triangles motivates the Pythagorean theorem (for the same material translated into Spanish see Pita'goras).
Symbols/Images is mathy.html.
Nelsen [9] has collected proofs (not motivations, but rigorous demonstrations) of facts. Two different ways to see equation (3) of [7] follow. Many people know that result as the product of n, (n + 1), and (2n + 1) divided by 6.


Sum Squares

Sum Squares By Three

The recent book [10] shows this and many more interesting items by an exposition both mathematical in style, and through visual means.
Mathematics evolved from dealing with practical problems. It is the basic knowledge that is essential to working with technology. [Periodicals' articles about modern technical issues and achievements, can start one in grasping the wonders of both. Some recognized sources are found from links at News.]
03/13/2014 Version http://www.cs.ucla.edu/~klinger/nmath/index2.html
©2011 Allen Klinger