The Square, the Circle and the Golden Proportion A New Class of Geometrical Constructions

Year: 2000 Authors: Janusz Kapusta

Core claim

The square-circle relationship yields new geometrical constructions and visual proofs involving the golden mean, powers of Phi, and approximate compass-and-straightedge subdivisions.

Topics

golden proportion, square-circle geometry, geometrical construction, visual proof

Domains

Euclidean geometry, golden ratio, Pythagorean theorem, compass and straight-edge construction, geometric art, visual pattern design, picture essay

Methods

diagrammatic exploration, geometric construction, sequence analysis, approximation

Media

squares, circles, triangles, gridded 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.

BRIDGES Mathematical Connections in Art, Music, and Science

The Square, the Circle and the Golden Proportion A New Class of Geometrical Constructions

Janusz Kapusta 1060 Ocean Avenue, Apt. D5 Brooklyn, NY 11226, U.S.A. E-mail: kapusta@earthlink.net

Introduction

The reason behind taking another look at the number Phi is its overwhelming appearance in art, nature and mathematics [1,2,3]. I feel that such a power must have a deep basis. As a result of this investigation I have discovered a new world of geometrical relationships residing within the square and the circle. This picture essay can be read as an example of how complexity emerges inexorably from simplicity.

Figure 1a and 1b: The square, the circle and 8 (√5/2) diagonals forming eight pointed star. Notice how the 10 x 10 grid appears naturally from it. The shaded square in the middle has side (1√5). When rotated so as to be vertical, the golden proportion appears as shown to the right and above. Many properties of this star have been investigated by T. Brunes and J. Kappraff [4,5].

Janusz Kapusta

Figure 2: A square.

Figure 3: 10 squares.

Figure 4: 2 tangent lines to 10 squares.

Figure 5: Appearance of a new square with upward pointed triangle.

The Square, the Circle and the Golden Proportion – A New Class of Geometrical Constructions 249

Figure 6: Upward, downward and sidewards triangles form an 8-pointed star.

Figure 7: The 8-pointed star is expanded to a nine-square grid.

Figure 8: Notice how the 8-pointed star is related to the original sequence of squares.

Janusz Kapusta

Figure 9: Circles are placed within the squares.

Figure 10: A triangle is formed tangent to the circles from which a pair of circles are defined with diameters in the golden proportion. Figure 11: The upper square is seen in exploded view.

The Square, the Circle and the Golden Proportion – A New Class of Geometrical Constructions

Figure 12: A sequence of “kissing” (tangent) circles are created with the negative powers of the golden mean.

Janusz Kapusta

Figure 13: A visual proof that the odd negative powers of the golden mean sum to unity [5].

Figure 14: All the negative integer powers of the golden mean with the exception of $1/\varphi$ sum to unity.

The Square, the Circle and the Golden Proportion – A New Class of Geometrical Constructions 253

Figure 15: Another way to view the odd negative powers of the golden mean as a sequence of circles.

Figure 17: The other infinite sequence is seen as geometric series of squares and circles of decreasing size.

Figure 16: They can also be seen as a sequence of squares.

Figure 18: The Pythagorean theorem is expressed by this sequence of squares. Notice how a sequence of vertices of the squares upon the hypotenuse lie against the right edge of the framing square.

Janusz Kapusta

Figure 19 and 20: An approximate compass and straight-edge construction of the angle of $3/56\times 360$ degrees can be related to the golden mean. I found the error to be $0.12\%$ . Since $7/56\times 360$ degrees equals 45 deg. this means that the circle can be subdivided by compass and straight-edge into 56 equal angles to close approximation. It should be noted that the Aubrey circle at Stonehenge has 56 equally placed stones [6]. The 56 subdivisions enable a heptagon to be constructed with compass and straight-edge to within $0.73\%$ error.

Figure 21 and 22: As a result of these findings I have come upon two new constructions of the golden mean based on the relationship between the circle and the square.

Acknowledgment

I would like to thank Professor Jay Kappraff who showed great interest in my search and not only encouraged me to present part of the findings in this paper but helped in its editing.

Bibliography

1. Kappraff, J. Connections: The Geometric Bridge between Art and Science. New York: McGraw-Hill (1991)
2. Hertz-Fischler, R. A Mathematical History of the Golden Number. Mineola, New York: Dover Publications (1998)
3. Huntley, H.E. The Divine Proportion: A Study in A Mathematical Beauty. New York: Dover Publications (1970)
4. Brunes, T. The Secrets of Ancient Geometry and its Use. Copenhagen: Rhodos (1967)
5. Kappraff, J. Beyond Measure: A Guided Tour through Nature, Myth, and Mathematics. (In press)
6. Hawkins, B. Stonehenge Decoded. New York: Dell (1966)