The trisection of an angle is a famous problem used to encourage innovative thinking. However recently , since the algebraic proof of impossibility, it has been used to brainwash vulnerable mathematicians into a hopeless conformity.
The solution was clearly found by the ancient Sumerian and Akkadian peoples , the Dravidian and Harrapan Indus Valley civilisations and the Mongol chinese steppe and Plain civilisations, all of whom had the wheel and the 60 modulo arithmetics.
The issue is a pragmatic metrical one, and relies on skillful Neusis, as well as expertise with circles
Such an expertise is now called sacred geometry, but it is a science of spherical and circular relations. Of all the forms we have explored it is the circle that encodes proportion in its simplest form: one perimeter to 1 diameter!
We all accept that a circle can be patterned by six petals formed by 6 overlapping circles. We accept the number 6 because we see symmetry. This means we cannot distinguish the 6 forms we see in the pattern by any known or used measurement; accept by calculus! In that branch of ” precision” we find pi to be not 3 but 3.1415… Because our calculation is not based on observation but by a division process!
The difference is profound. Do we trust our eyes or our formal calculation process? Both, because as it turns out our ancestors did not need precision. 6 was good enough for them even though we know that it should be 6.28…
When your compasses do not quite meet the diameter we are taught to do it again until it does, because the radius must step 6 times into the circumference! It does not, but by convention we say it does.
The pattern of 6 is so compelling , we want 6 equilateral triangles as a constructed Constant of space. Construct them in a circle and they fit, construct them in a tessellation and they fit, but they do not precisely fit a circle anymore! The construction in the circle has slightly distorted the plane forms.
Using a constant radius we can construct the sacred geometrical flower pattern. That is when we can start to set out proportions . While we ca crowd 6 around a centre of a circle with 1 radius we can crowd 9×6 around a circle with 3 times the radius! This means we can trisect the diamond made of 2 equilateral triangles . But we have to use the chord length of the 1/3 rd circle to step these off on the larger( 3 x ) circle.
This proportion exists in this set up because circles are proportions. Without a rigid measure it is fiddly to do it is much simpler with a set of measuring tools that can retain and transfer these lengths to the proper positions
There exists a circle for which this length is the precise chord which trisect the arc into 3 similar sectors. Finding it by trial and error can be made easier by using the sacred geometry to narrow the search down. The Neusis becomes simpler and more precise.
Draw an angle and make the limbs or rays long enough to step off 3 radii. The radius is the semi circle drawn at the vertex of the angle, extend your compass to 3 radii and draw the semi circle,
Using a pair of rigid divider measure the chord of the angle in the smaller semi circle.
Step this off on the larger arc until step 3( which is too small ) and step 4 (which is too large).
Leaving the dividers fixed , now use the intersection of the upper ray between steps 3 and 4 with the semi circle as the centre for a circle that has a radius given by the displacement to step 4. Retaining that radios go to step 3 and mark an intersect toward 4 as the centre of a second circle through 3
Now using the point of intersection of these circles with the ray draw a semi circle from the vertex. Using the dividers step off to point 2 along this arc. Setting your compass to the displacement from the point on the ray cut by this arc( the same as that cut by circles at 3 and 4) now draw a circle that intersects circles 3 and 4
The circles are a probability space . Where they intersect is probably the point for the radius of a semi circle which can be trisected by the dividers precisely.
There may be 2 intersections that are clear. Choose the one that fits best.
Where these circles cut the upper ray is a point which was used to draw a semi circle. That semi circle will be unable to contain more than two steps within the arc of the angle.
The demonstration relies on neusis so be as accurate as possible.
The empirical deduction is that the 3 x radius semi circle is going to present an arc( the angle) which being less curved will be too big, by a proportion . Points 3 and 4 are used to narrow the space that the sought for circular arc must pass through. By using the 4th point as a radius displacement and drawing a circle the bounds can be seen to decrease until the circle cuts the upper ray. Thus any circle drawn from the vertex passing through that circle has a high probability of approaching the required radius from above.
The second circle passing through point 3 from a marked centre has the probability of a circle approaching the correct circle from below. Thus both bound a circle which will likely contain 3 to 4 steps
The second circle will bound a circle that is likely to contain 2 to 3 steps
Where they intersect has a high probability of being the correct circle requiring precisely 3 steps to equal the angle.
Clearly the dividers must be kept rigid and the stepping off done as accurately as possible.
Should the result not be “perfect” then the two guiding semi circles, just drawn , can be used to repeat the method.
You will find that if the angle is a 60° or 90° or some multiple of those the circles at 3 and 4 will very nearly coincide. Do not neglect to differentiate the points of intersection.
At first I thought this was a method of approximation relying on proportions un related to sacred geometry, but when I saw that the circle count was 4:3 for the 120° I could see then the sacred geometrical pattern peeking through. The first radius would cut the smaller circle into 6, but the chord was cutting the 3 x circle into approximately 12. The circle I sought would be cut precisely into 9 by that same chord.
These are empirical findings, the sort every geometer should be looking for as a matter of professional expertise!