I bought a copy of TASCHEN's "The Elements of Euclid" with illustrations by Oliver Byrne from his book in 1847. As I was working through a lot of the later problems I could see myself causing these sorts of errors due to a skill issue. For example as I dragged the pencil vertically down a ruler the wooden side of the pencil (above the lead) rested on the ruler. As I dragged the pencil down my arm lowered and my wrist changed orientation so that the lead of the pencil was now resting against the ruler as I completed the line. It was honestly so frustrating to see the corners and arcs of my diagram look like abstract dog shit because I was too much of a dumbass to use a ruler correctly.
It seems to me that the placing of one of the points is “exact.” Unless you are working on graph paper or something (but that sort of goes against the spirit of classic geometry I think), there’s no absolute frame in which the first point can be incorrect, the rest of the construction is relative to it, right?<p>Also, is the inaccuracy around B actually a circle? I think you could easily end up missing that horizontal line he’s added between the imagined “right A” and “right B.” But, that line shouldn’t exist yet, it connects A and B wherever they are. The position of B is wherever your pencil lands.<p>The second circle seems to have more room for error. You can miss B with the pointy end, and could flex the compass differently from when you did the first circle, resulting in a different BA distance. Although, it should be possible to visually check this.<p>Is that right? I somehow never managed to take a geometry class.
It's not a bug, it's a feature. All students know that any line goes through two big enough points.<p>I was expecting something related to "exact predicates", like for example:<p>Efficient Exact Geometric Predicates for Delaunay Triangulations<p>Which is not "classical geometry", I guess, although it is constructible with ruler and compass (you need to draw circumcircles of triangles and maybe change triangles after that)
> In this sense, the Apollonius perpendicular bisector construction appears to be sensitive to the errors of compass placement.<p>It looks like errors are reasonably under control. It looks like the input error is a little over 5% of the distance AB and so it's not surprising that after a few steps, the output error in the position of C is around 15-20%. If you started with input error under 1%, the output error would be under 4%. There's no large error blowup here.
Hamkins has some great material, including his book "Lectures on the Philosophy of Mathematics". What's even better is that it is accompanied by a video lecture: <a href="https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIngyROBsUqcf_z" rel="nofollow">https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIn...</a><p>I am watching them while doing the dishes :-)
The writer is considering the subject as a physicist might. Euclid’s figures are ideal objects, they don’t have errors. It’s assumed that your construction is an approximation. Euclid might have drawn his figures on sand with lots of “errors”, still the figures are idealized no matter how you draw them. For instance, Euclid represents numbers as lines. There are not units and no errors. His proofs are still valid if I don’t draw his lines exactly as he drew them (I’m talking about Book IX where there are propositins about numbers).
What an answer to the problem of trisecting the angle!<p>If you figure your angular accuracy is 0.3 degree or something you can construct a binary fraction like 341/1024.
Welcome to laying out sheet metal:<p><a href="https://www.youtube.com/watch?v=9DypIDbAVTc" rel="nofollow">https://www.youtube.com/watch?v=9DypIDbAVTc</a>
I thought this is going to be about floating point inaccuracies. For example, here's a Blender Geometry Nodes setup that instead of constructing a circle using a provided node "Circle", or by positioning and connecting points, which coordinates are calculated independently using an incremented angle, sine and cosine, instead draws a circle by walking in a circle: on each step ("extrusion"), the direction of the step is rotated by 360°/total_steps (for 360 steps it's rotated by 1°). The more steps there are, the bigger the accumulated floating point error (the error is guaranteed when positioning a new vertex, which position is stored as a xyz triplet of float32 values, but I imagine in Blender code floats are used as well).<p>video: <a href="https://streamable.com/0xba64" rel="nofollow">https://streamable.com/0xba64</a><p>.gif: <a href="https://i.imgur.com/4Y33nrf.gif" rel="nofollow">https://i.imgur.com/4Y33nrf.gif</a><p>.blend file: <a href="https://e.pcloud.link/publink/show?code=XZvFUnZKrqbyl9h8ab3UCrJxdlBLHaW6qsX" rel="nofollow">https://e.pcloud.link/publink/show?code=XZvFUnZKrqbyl9h8ab3U...</a><p>Going back to the article, I like the illustrations depicting all possible values as areas; it seems you could calculate with a formula the boundaries of those areas and so draw them efficiently and allow to dynamically modify precision, ciechanow.ski style...