Improving Access to Fundamental Concepts
Allen Klinger

Synopsis

This paper concerns using the ability of inter-net computer communications software and hardware to connect people to practical problems. Current technologies support disseminating visual information, sounds, animations, as well as text files. Pedagogically those methods enable improvements to engage people in abstract ideas where currently many obtain only an acquaintance with related symbols.

The heart of this article involves three visuals, a lunar image, a three-dimensional solid with hexagonal-equator composed of rectangular and triangular faces, and a series of nested circles with polygons inscribed (the largest polygon is a triangle) with successively higher numbers of edges. All three images can also be seen easily by using an inter-net browser (Netscape Navigator, Internet Explorer, Safari, Mozilla, etc.); their URLs are in the text. The image of circles with inscribed polygons is from a book by Kasner and Newman, first published in 1940; those authors demonstrate there that ancient Greeks knew about limits and accurately evaluated the limit value of inscribed circles’ radii.

The paper concludes with descriptions of, and pointers to, thought-provoking topics. Each topic is two things. On one hand each challenges the reader, and requires work. But each stimulates review of some basic notion. The text suggests some connections to fundamental concepts and useful tools.

Introduction

Both engineering students and people in the general community can benefit from accessing inter-net materials that explain abstract ideas. These materials can be text files, but also hypertext (links to related topics), images, animations, and sounds. Because practical problems deal with real situations, visual methods often lead to, or motivate solutions.

Visual and animated material fix ideas in one’s mind; they make concepts memorable. In general visual items make the material more understandable and hence accessible to more people. The following three examples are two- and three-dimensional objects with the latter shown through a planar image generated by computer graphics software (all also can be viewed from their URLs). One portrays a lunar image. The next is a three-dimensional solid composed of rectangular and triangular faces (and a hexagon for its equatorial edges). The last one is a series of nested circles with polygons inscribed (the largest one is a triangle). The polygons with successively higher numbers of edges go inside the circles of shrinking radii.

http://www.cs.ucla.edu/~klinger/nmath/2circles_9_13_04.pdf

Figure 1. Lunar Image Constructed by Circles

http://www.cs.ucla.edu/~klinger/tet1w.gif

Figure 2. Solid Constructed from Vertices’ Coordinates

http://www.cs.ucla.edu/~klinger/radiia.jpg

Figure 3. Image With Polygons Inscribed in Circles [1]

Partitions

Three questions each deal differently with physical division. The first is quickly stated in words: “can a circular pizza be divided into eight equal portions with exactly three straight line cuts?” (Many find this difficult.) Addition, subtraction, and partitioning an entity into fractions, are commonly understood, yet can lead to difficulty with: “can you combine the four digits 1, 5, 5, 5, using only the arithmetic operations: plus +, minus -, times x, and divide, possibly using parentheses; to obtain the value 24?” The third is an ancient question. It is “can you divide nine loaves of bread among ten men so that each has a fair share?” [In addition to being equal in quantity the portions should seeming to be the same, as much as possible.] It too asks about applying fundamental knowledge to a practical end.

Answers to all three also appear at or are reachable from web URLs. The following two figures show some visual aspects of these questions. [Regarding the second figure, additional material about the modern and ancient Egyptian views of fair division appears at http://www.cs.ucla.edu/~klinger/efractions.html. People seeking an animated statement of the 24-question, with a clickable button to reach the solution, may consult http://www.cs.ucla.edu/~klinger/nmath/puzzle1555.html.]

http://www.cs.ucla.edu/~klinger/pizza1.jpg

Figure 4. Pizza Image With Text [2]

http://www.cs.ucla.edu/~klinger/eimage.jpg

Figure 5. Modern and Ancient Egyptian Views of Equal

Whether the “24 from 1, 5, 5, 5” question answer comes easily depends on how comfortable one is with fractions. The irreverent image in Figure 6 and delays in getting a 24 answer reveal that this is an uncomfortable subject.

http://www.cs.ucla.edu/~klinger/19-95c.jpg

Figure 6. Not The Best Way to Cancel

Internet Access to Images

Only an elementary level of skill came into the creation of Figures 1, 3, and 5. Some knowledge of lists and the graphics behind Mathematica images supported Figure 2. Figure 4 was by someone with design and art skills. The URL in Figure 6 leads to an animation. All involve (see acknowledgement) interaction with students, colleagues, and books. While these files demonstrate the power available from using computer graphics and animation along with text, to present material via the inter-net, much more sophisticated still visuals, dynamic, and evolving materials are accessible.

First, consider the beauty and insight conveyed by the URLs in Table 1. The material there concerns just two things: a) turning a sphere inside out; and b) sixty-three famous curves. The first two URLs play movies showing the inside-out transition. The last URL lets one load programs that enable selection of the parameters of the curve in question.

On Turning a Sphere Inside Out

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

http://www.geom.uiuc.edu/docs/outreach/oi/evert.qt

http://www.akpeters.com/product.asp?ProdCode=2183

http://www.xs4all.nl/~alife/sphere1.htm

Links to Parametrically Changeable Versions of Sixty-Three Famous Curves

http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html

Table 1. URLs of Material Available From the Inter-Net

Second, although the Figure 1 image may be familiar from looking at the night sky, embedding that knowledge about overlapping circles in a source statement (“familiar from looking at the night sky”), is itself not part of every course covering geometry. Likewise other practical (or discovery and origin aspects) aren’t always available. But an embedding of that image with related materials is now accomplished (although not by someone who interacts with students in K-12 education) via:

http://www.cs.ucla.edu/~klinger/nmath/ellipse.html

This material connects overlapping circles to squares and ellipses. It does this both by written text and by pointers (links, hypertext).

Conclusion

Perhaps you now believe that the pizza cutting let you see more in 2³ = 8 than you had before. Possibly the fourteen faces in tet1.jpg and tet1w.gif seem a good display of twelve points equidistant from the origin, six in the equatorial plane, three at each pole. But almost surely you are now in possession of ways to see sixty-three famous curves, and watch a sphere turn inside out. All that makes you a candidate for advertising how the inter-net improves access to mathematical knowledge.

References

[1] Kasner, Edward and Newman, James R., Mathematics and the Imagination, NY: Simon and Schuster, 1940; Redmond: WA, Tempus Books of Microsoft Press, 1989; pp. 310-312

[2] Gillings, Richard J., Mathematics in the Time of the Pharaohs, Cambridge, MA: The MIT Press, 1972; NY: Dover Publications, Inc., 1982, ISBN 0-486-24315-X.

Acknowledgement

Gavin Wu created the image labeled pizza1.jpg that appears in Figure 2.

Yu-Chian Tseng created the animation labeled puzzle1555.html .

Walter Karplus called my attention to the Figure 6 cancellation.

Eskandar Ensafi and unknown Wolfram Research employees assisted in the creation of tet1.jpg and tet1w.gif. Dorene Lau created material at dorene/math3.htm that was excerpted for nmath/ellipse.html .