1. Straight Edge and Compass Euclidean Constructions
From the time of ancient Greece, geometers have been fascinated by what can be built using only two tools: a compass and a straightedge. These "Euclidean constructions" formed the foundation of classical geometry. You can copy a line segment, bisect an angle, draw perpendiculars Euclid's Elements are full of constructions and theorems that are built from these two tools.
But some polygons cannot be drawn this way. A regular 7-gon is impossible. Why? Because the set of constructible polygons is tied deeply to number theory, to Fermat primes.
2. Fermat Primes and Drawing Them
A Fermat prime is a prime number of the form
Only five such primes are known: 3, 5, 17, 257, 65537
Carl Friedrich Gauss at age 19 by proved that a regular polygon with sides is constructible if and only if is a product of a power of 2 and distinct Fermat primes. This means a regular 65537-gon is theoretically constructible.
One mathematician actually attempted to do it. Johann Gustav Hermes spent over a decade and 200 pages , detailing how to construct a 65537-gon using Euclidean methods in Ueber die Teilung des Kreises in 65537 gleiche Teile in 1894.
3. The 65537-gon
If you tried to actually draw a 65537-gon on a circle the size of the Earth (with of 6,371 km), each side would be approximately 611 meters long. Which means on a flat lake one side of the 65537-gon would not follow the earth's curvature by 3cm. A 65537-gon would be 3cm per 611 meter side off being a perfect circle.
No one has ever made a by hand 65537-gon because it would just be too involved. though Hermes did spend ten years working out how it could be done in principle.
4. A Machine to Draw It for Us?
Could we build a machine that uses only a compass and a straightedge, like Euclid himself? Plotters could draw the 65537-gon by calculating angles and coordinates, but Euclid has famously tough lawyers and anyone making his shape not using his methods is likely to be mired in law suits
Such a Euclid Geometry machine would have to:
Set distances using a mechanical compass
Draw arcs
Align a straightedge through marked points
Draw straight lines with a pen
Combine these steps tens of thousands of times
It would follow the ancient Greek rules of construction.
This would be rock hard to make and not practically useful. It would involve, aligning a ruler through two points. Resetting a compass to exactly the distance between two previous marks, which is mechanically fiddly. Detecting intersections and points on paper. All of this is simple for Ancient Greek humans but not for machines.
And yet, it would be beautiful. It would make visible not just the result of geometric thinking, but the process. A Euclid-bot wouldn’t just draw geometry it would demonstrate it.
There are cool online geometry tools but something about plotters and physical drawing things still appeals. Maybe it would be possible to make a ruler and compass wielding machine that could finally draw the 65537 sides of the last Polygon.