99 Points of Intersection: Examples-pictures-proofs by Hans Walser

By Hans Walser

Translated from the unique German via Peter Hilton and Jean Pedersen

The ninety nine issues of intersection awarded right here have been accumulated in the course of a year-long look for fantastic concurrence of traces. for every instance we discover compelling facts for the occasionally startling proven fact that in a geometrical determine 3 immediately traces, or occasionally circles, go through one and an identical aspect. in fact, we're conversant in a few examples of this from simple ordinary geometry — the intersection of medians, altitudes, perspective bisectors, and perpendicular bisectors of aspects of a triangle. right here there are numerous extra examples — a few for figures except triangles, a few the place much more than 3 directly strains go through a typical aspect.

The major a part of the e-book offers ninety nine issues of intersection in simple terms visually. they're constructed in a series of figures, many with no caption or verbal observation. furthermore the e-book includes common ideas on and examples of the issues of intersection, in addition to a few average equipment of proving their life. the various examples proven within the ebook have been encouraged through questions and proposals made through scholars and high-school lecturers. numerous of these examples haven't just a geometrical, but additionally an exciting aesthetic, element.

The ebook addresses high-school scholars and scholars on the undergraduate point in addition to their lecturers, yet will attract someone attracted to geometry.

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Origami, or paper folding, may be viewed as a curious branch of geometry. On January 26, 1887, Carroll mentions that he folded a fishing boat (it had a seat at each end and a [27] THE UNIVERSE IN A HANDKERCHIEF basket for fish in the middle) and a paper pistol for a little girl. On November 16,1891, he writes of teaching the children of the Duchess of Albana how to "blot their names in creased paper," and how to fold paper pistols. The paper pistol is a ® *M' ^^i^**^' @ / f How to fold paper pistols.

You can see this demonstrated by an exhibit in Chicago's Museum of Science and Industry. " There is no evidence that Carroll ever followed up this suggestion. In 1991 I tried to invent the kind of game Carroll had in mind. It is currently on sale as Wordplay, availablefix)mKadon Enterprises, 1227 Lorene Drive, Pasadena, Maryland 21122. On March 8,1887, we find this entry: "Discovered a Rule for finding the day of the week for any given day of the month. " Carroll published this rule as a short note in Nature (March 31, 1887).

Healthy Englishmen. B. Hermits. C. Strong hermits. D. Strong people. E. Healthy people. NoAisB. AllCisE. AUAisD. AllCisB. AllAisE. A11 C isD. I analyzed the problem by shading regions on a Venn diagram for five terms. All regions where the set of healthy Englishmen overlap the other sets are eliminated but one, where A overlaps E and D, showing that all healthy Englishmen are healthy as well as strong—facts that we already know from the first three premisses.

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