September 30, 2024. © Copyright 2024 by Paul Glossop. All rights reserved. For more information, please email: Geometer On to the Hendecagon! On to the Hendecagon!
another division of Geometer
Constructing the Nonagon
Regarding the nonagon and indeed all regular polygons, please realize that prior to about 100 years ago, there wasn't such a concern with mathematical perfection as there is today. For our forebears, a variance from the ideal of 1 part in 1000 was not discernable, and therefore the solution was regarded as excellent. A minimum of construction steps was also very important. The ideal line is of zero thickness, and yet our visible geometric lines are far from the ideal. At each step of Euclidean construction (straight edge and compasses) a small error occurs due to the limits of the human eye and the thickness of the line. With more steps of construction errors accumulate, and the aggregate error almost certainly exceeds the theoretical. And yet why were the polygons, particularly the irregular polygons, of any concern at all in history? They were the 'pure mathematics' of the day, easily imagined but challenging to construct. Also they were approachable by all, professional mathematicians and amateurs alike. Most of polygons, especially the odd-numbered ones, were not particularly useful in practical areas such as architecture. The next odd polygon, known and the hendecagon or undecagon, is eleven-sided. It is very simple to find the basis for its key angle. v = 0.00154° v = –0.01235° Variance of Pink angles  (+) 4 parts in 100,000 Ideal Angle ( ) Pink angles ( ) Ideal Angle  ∙105 = –3.85877 Variance of Key angle  (-) 3 parts in 10,000 Ideal Angle ( ) Key Green angle ( ) Ideal Angle  ∙104 = 3.08702 Ideal Angle = 40.00000° A n a l y s i s Pink angles = 40.00154° Key Green angle = 39.98765° Suppress construction lines and draw the facet lines of the nonagon. The 8 pink triangles subtend equal angles at centre O. The key, green triangle subtends an angle at O that is very slightly different. J D C B H G N Y E O C1 C1 C1 C1 Construct bisectors of GOY, GON, HON, HOE to intersect C1 in B, C, D, J respectively Construct bisectors of NOY and NOE to intersect C1 in G and H respectively Find midpoint M of YE. Produce MO to intersect C1 in N. Suppress construction lines, leaving the key angle. J D C B H G N Y E O H G N Y E O N M Y E O Y E O C3 C2 C1 9∙mYOE = 359.88887° 360° 9 = 40.00000° mYOE = 39.98765° Constructing the Key to a Nonagon 1.) Presume the founding circle C1 and its circumsquare. 2.)  With centre V and radius VO, draw circle C2 to intersect square in D. 3.) With centre D and radius DO, draw circle C3 to intersect circle C1 in Y. 4.) Angle YOE is very close to the angle in a nonagon. Y D V E O This Nonagon-based Mandala was constructed using the steps shown below. When the nine triangles that comprise the inner nonagon are pointed to the centre, the nonagon rests in a state that represents undivided, whole knowledge. When the same triangles are 'hinged 180° outwards', then the nonagon is seen in its most expressive state. This helps us to understand the nature of knowledge: it has inner wholeness and coherence, and yet can express itself in multiple points of diversity.