Mathematical Secrets of Seven

Year: 2008 Authors: Susan McBurney

Core claim

Seven produces useful geometric and arithmetic patterns, including approximate heptagon constructions and distinctive divisibility and repeating-decimal behavior.

Topics

regular heptagon, approximate construction, repeating decimals, divisibility by seven, ornamentation

Domains

Euclidean geometry, polygon measurement, number theory, modular patterns, ornamentation, architecture, mathematical design, pattern-making

Methods

straightedge and compass approximations, angle calculation, area derivation, divisibility tests

Media

circle, square, line segments, diagrams

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Mathematical Secrets of Seven

Susan McBurney Western Springs, IL USA E-mail: smcburne@iit.edu

Abstract

The number seven is associated with a variety of phenomena, both real and symbolic. However, in the physical world of ornamentation and architecture, it is less widely utilized than other numbers. This paper examines the regular heptagon in some depth and looks at particular characteristics of the number seven itself.

Nearly everyone agrees that all numbers are created equal, that numbers are just numbers, and nothing else. They have no inherent special meaning, and yet they can be used to describe the workings of the universe. Such is the beauty of the science of mathematics.

The fascinating worlds of architecture and ornamentation often make use of regular figures, particularly regular polygons which can be constructed easily and accurately. There is a preference for those figures with even numbers of sides. On the two-dimensional plane, tiling often figures prominently and of course only the triangle, the square, and the hexagon can cover the plane without gaps or overlaps.

Even so, the pentagon appears fairly often with it’s association with the Golden Section and a perceived inherent attractiveness. But for a variety of reasons, seven-sided figures do not appear as often in ornamentation as other regular-sided polygons. Even the enneagon (9-sides) is more common although it, too, is relatively rare.

Approximate Construction of Regular Heptagons

The regular heptagon cannot be precisely constructed using only a straight-edge and compass. However, a number of very close approximate constructions have been devised, some of them dating back to antiquity. Figure 1 catalogs five of these which will be explained briefly here. [2], [4], [5]

Logically, by calculating $360/\mathrm{n}$, we see that the central angle of the heptagon should be 51.428571… degrees. We can determine the accuracy of each approximate method by comparing it’s central angle with this number.

a) Draw the radius of the circumcircle. Construct a segment of the same length perpendicular to the radius. Find the point E such that CD touches the circle. OE will be the length of the side of the regular heptagon. Notice, this construction requires a measurement, or a marked straight-edge and is therefore not a classic Euclidean construction.

b) Given an equilateral triangle within the circumcircle, find the midpoint C of one side. Using this length, scribe an arc from the vertex F of the triangle to cross the circle at G. The chord FG will be the length of the side of the regular heptagon. This method was used by Albrecht Dürer in the early $16^{\text{th}}$ century.

c) The angle whose tangent is $5/4$ ($51.3402^{\circ }$) is very close to the central angle of the regular heptagon.

d) Beginning with a circle, construct a square enclosing it such that the sides are tangent to the circle. From two adjacent corners, draw quarter circles with the radius equal to one side of the square. Connect the point of intersection of these two quarter circles (I) to one of the corners.

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Figure 1: Methods for approximate construction of regular heptagons

Call the intersection of this line with the circle, F. Draw a segment from F to the midpoint of the base of the square, G. FG will be the length of the side of a regular inscribed heptagon.

e) Beginning with six arcs drawn to divide the circle into six equal parts, draw an arc from one point that “just touches” the neighboring petals and continues to cross the original circle. At the crossing point F draw a line to the tip of the second distant petal. Bisect this double angle FOH to find point G. FG will be the length of the side of the regular heptagon.

Measurements of a Regular Heptagon

Having constructed an approximate regular heptagon, what do we know about the actual measurements contained within it other than the central angle of 51.428571 degrees?

The vertex angle can be deduced from the following: Draw seven interior triangles. Each triangle has a total of $180^{\circ }$ so there is a total of $7(180)=1260^{\circ }$ . Now subtract the central angles $(360^{\circ })$ to leave just the total of the vertex angles. All the vertex angles are equal, so they must each be $900/7$ or $128.57142^{\circ }$ .

The radii of the circumcircle and the inscribed circle can be calculated in terms of the length of the side E of the polygon. See the diagram in Figure 2. In that figure a close look at $\frac{1}{2}$ the central angle (ABC) gives an equation for finding the length of the radii.

Figure 2: Finding the radii of the incircle and circumcircle of any regular polygon

The incircle radius (r) of a regular heptagon is e cot $(180/7)=(\mathrm{E}/2)$ $(2.0765)=1.0383\mathrm{E}$ (the length of one edge). The circumcircle radius (R) is e csc $(180/7)=(\mathrm{E}/2)$ $(2.3048)=1.1524\mathrm{E}$ .

To find the area of the interior of the polygon, first find the area of $\Delta ABC$ , double it, and multiply by n. For the heptagon, the area of $\Delta ABC=\frac{1}{2}$ (base)(height) = $\frac{1}{2}$ e e cot (180/7). Doubling this and multiplying by 7 gives $7{\mathrm{e}}^2$ cot(25.7143) = 14.5356 (E/2) ${}^2$ = 3.6339 E ${}^2$ = area of regular heptagon where E is the edge length.

Repeating Decimals and Divisibility by Seven Tests

Moving from geometry to number analysis, we see that an interesting pattern develops when dividing by seven. Division of any number by seven yields a repeating string of the same digits, 142857… in the same order, except in cases where the number is a multiple of seven. Three, 6, and 9 never occur.

There is a rather unusual, if inefficient, test that will determine if a number is evenly divisible by seven. Extract the last (ones) digit, multiply it by 5, and add it to the remaining digits to get a new number. Now repeat the test if necessary, until you arrive at a small number which you can determine is or is not a multiple of seven. [3] Here is an example:

945,238; 94,523 + 5x8 = 94,563; 9,456 + 5x3 = 9,471; 947 + 5 = 952; 95 + 5x2 = 105

$10+25=35$ which is evenly divisible by 7!

Here is a further technique for testing very large numbers. Divide the number into three digit clusters (as by grouping with commas). Take the first (left-most) cluster, subtract the second cluster, to the result add the third cluster, subtract the forth, and so on. When you are finished if you still have a large number, you can apply the previous technique. By the way, this test also works for the numbers 11 and 13.

Consider the number 784,639,801,295.

$784-639=145145+801=946946-295=651$ which is evenly divisible by 7.

Conclusion

Using the characteristics of seven in an artistic manner provides a special kind of challenge to would-be designers, but the repeatability of digits after division offers one good place to start. Here are two designs using lines of lengths in the same order as the remainder digits 142857… The square offers interesting interior spaces which emphasize the pattern of the digits.

Figure 3: Two designs implementing the remainder digits 142857 after division by 7

References

[1] Robert Williams, The Geometrical Foundation of Natural Structure, Dover, pp. 32-33. 1979 [2] Robert Dixon, Mathographics, Dover, pp. 34-35, pp. 39-40. 1987 [3] Ravi Vakil, A Mathematical Mosaic, Brendan Kelly Publishing, Inc., pp. 27-29. 1996 [4] Michael S. Schneider, A Beginner’s Guide to Constructing the Universe, Harper Perennial, p 227, p 229-230. 1995 [5] Paul A. Calter, Squaring the Circle, Key College Publishing, pp. 147-148. 2008