Constructing the Heptagon
a division of Geometer
The Heptagon is the first regular polygon that is challenging to construct using straight edge and compasses, the standard Euclidean method. A newly-discovered construction enables us to get very close to the ideal. Shown below we start with the union of circle and circumsquare. Then in just three steps, we find the key. Using the key we complete a heptagon whose interior angle variances range from 7 parts in 1,000,000 to 8 parts in 10,000,000. C3 C2 C1 360° 7 = 51.42857° 7∙mHOE = 359.99760° mHOE = 51.42823° Constructing the Key to a Heptagon 1.) Presume the founding circle C1 and its circumsquare. 2.)  With centre V and radius VO, draw circle C2 to intersect square in B. 3.) Draw line segment BR to intersect circle C1 in L. 4.) With centre L and radius LO, draw circle C3 to intersect circle C1 in H. 5.) Angle HOE is delightfully close to the angle in a heptagon. H L B R V E O C1 C1 C1 C1 Using the Key, Complete the Heptagon: Bisect the REFLEX angle AOH to intersect C1 in P. AOP = HOP Bisect the REFLEX angle AOE to intersect C1 in N. AON = EON Find midpoint M of HE, Produce MO to intersect C1 in A Suppress construction lines P N A M H E O N A M H E O A M H E O H E O September 30, 2024. © Copyright 2024 by Paul Glossop. All rights reserved. For more information, please email: Geometer C1 Why do we now, and the ancient Greeks before us, engage with regular polygons? I believe we are seeking to discover and then to emulate how nature breaks wholeness, here represented by the circle, into symmetries and equal parts. An example is the clematis flower, with seven petals. How does nature's intelligence produce this particular seven-ness -- surely a deeper method than ruler and compasses? p.g. v = –0.00004° v = 0.00026° v = –0.00034° Variance  (-) 8 parts in 10,000,000 360° 7 mGON 360° 7  ∙107 = 8.31797 Variance  (+) 5 parts in 1,000,000 360° 7 mEOP 360° 7  ∙106 = –4.99078 Variance  (-) 7 parts in 1,000,000 360° 7 mHOE 360° 7  ∙106 = 6.65438 Group 1 HOE = 51.42823° Group 2 EOP = NOH = 51.42883° Group 3 GON = AOG = TOA = POT = 51.42853° 360° 7 = 51.42857° mPOT = 51.42853° mTOA = 51.42853° mAOG = 51.42853° mGON = 51.42853° mNOH = 51.42883° mEOP = 51.42883° mHOE = 51.42823° A n a l y s i s Suppress construction lines. Construct the perimeter of the heptagon. Triangles of similar colour subtend equal centre angles. Bisect NOA and POA to intersect C1 in G and T respectively. T G P N A H E O T G P N A M H E O On to the Nonagon! On to the Nonagon! On to the Nonagon! On to the Nonagon!