Magnified Details of the 60° Trisection— I found a way to trisect a 60° angle with Euclidean construction simulated on a computer using the GeoGebra Geometry Calculator . The software allows a user to zoom into details and manage minuscule structures, acceptable according to Euclidean principles. And GeoGebra's drawing board can be configured to display Cartesian coordinates, which are defined as centimeters in the following examples to provide a sense of scale. The construction employs a sequence of three progressively smaller linear thirds to geometrically bridge the length disparity between one third of the primary angle's chord and one third of its arc.  With the 60° primary angle having a base width and arc radius of 14 cm, the construction culminates with operations executed within a cluster of three intersections the size of a single bacterium, 2.66 µm x 0.923 µm. This grouping is so tiny that if it were constructed using the finest pen on a sheet of pape...

60° TRISECTION MADE POSSIBLE.  The chasm between 60° and 20° angles was thought to be so wide that after two millennia of failing to bridge it, euclidean angle trisection was finally written off by mathematical theory back in the 1830s. The claim was that trisection had been solved by proving it was mathematically impossible. But many trisection diehards soldiered on, and those who assailed mathematicians with drawings were routinely dismissed out-of-hand. Despite its deep influence on mathematics, no one would take the question of trisection seriously again.  I became one of those diehards. I couldn't agree that even the most rigorous mathematics could account for all the variables or prove impossibility. So I toiled on the problem and thought I'd nailed it a few years ago, but was off by a few hundredths of a degree after testing with better software. Through the process, I'd grown to appreciate geometry and persisted, this time using GeoGebra's Geometry Calculator. N...