7 ms·

The propagation of error in classical geometry constructions (2018)

88 points by downboots 3y ago

throwaway81523 3y ago
2018
- dang 3y ago
  Added. Thanks!
deleted 3y ago
[deleted]
Sniffnoy 3y ago
Does anyone know where some of these follow-up posts might be?
- bmacho 3y ago
  > Unfortunately, I seem to have lost track of the references I had in mind. If anyone else can post suitable references, I’d be grateful. (2022)
konstantinua00 3y ago
[flagged]
justforasingle 3y ago
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.
- gilleain 3y ago
  Yes, I've learned some techniques after reproducing a lot of Islamic patterns, which can be done entirely with straightedge and compass (although frankly I often 'cheat' by using measures, grid paper and tracing paper, lol). Indeed the movement of the pencil can matter a lot, as can how good your compasses are. I've found that a bevelled ruler can be better if turned over so that the pencil rests against the bevel. Otherwise you have to take account of the width of the pencil lead and adjust the ruler position to take account of it. Certainly small errors in line position, length and (especially) angle can have a huge effect over a large diagram that has a lot of symmetries. It's surprising how noticeable even a small error of 0.5 mm between two lines is that should be colinear.
- mbivert 3y ago
  For that kind of things, lead-holders are preferable: you at least get a consistent lead width, despite the wearing, assuming you maintain the orientation of the pencil. An alternative is to sharpen your pencils with a utility knife & sandpaper: it's more versatile and less wasteful than a regular sharpener. For example, you could sharpen it to an edge (like a slotted screwdriver), and turn the pencil once in a while, so that the wearing helps preserving the edge. I've also dabbled with Islamic patterns recently. Side-note: I've found some high-quality, cheap second-hand compass sets, that used to be used for wood work a few decades ago (not made-in-China).
bee_rider 3y ago
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? 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. 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. Is that right? I somehow never managed to take a geometry class.
- deleted 3y ago
  [deleted]
- Warwolt 3y ago
  I think the circles are for when you create the arc using the compass. Presumably you already constructed points A and B and joined them into a segment, so there's now a point B to miss.
  - bee_rider 3y ago
    If you are just trying to show the relationship between the lines, I don’t think there’s any reason to set A and B beforehand. Of course, it could be the case that there’s a multi-step proof where you do care about the positions of A and B in relation to some other things. But I think that would require doing the analysis with that in mind. Maybe the order in which you construct the more complicated proof can also benefit from picking certain points to be exact.
- simplicio 3y ago
  I think the construction is usually phrased: "given a line, to..", so it assumes a pre-existing line segment, rather then that being part of the construction itself.
bsder 3y ago
Welcome to laying out sheet metal: https://www.youtube.com/watch?v=9DypIDbAVTc https://www.youtube.com/watch?v=9DypIDbAVTc
adalacelove 3y ago
It's not a bug, it's a feature. All students know that any line goes through two big enough points. I was expecting something related to "exact predicates", like for example: Efficient Exact Geometric Predicates for Delaunay Triangulations 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)
auggierose 3y ago
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: https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIngyROBsUqcf_z https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIn... I am watching them while doing the dishes :-)
Etherlord87 3y ago
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). video: https://streamable.com/0xba64 https://streamable.com/0xba64 .gif: https://i.imgur.com/4Y33nrf.gif https://i.imgur.com/4Y33nrf.gif .blend file: https://e.pcloud.link/publink/show?code=XZvFUnZKrqbyl9h8ab3UCrJxdlBLHaW6qsX https://e.pcloud.link/publink/show?code=XZvFUnZKrqbyl9h8ab3U... 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...
- deleted 3y ago
  [deleted]
tromp 3y ago
> In this sense, the Apollonius perpendicular bisector construction appears to be sensitive to the errors of compass placement. 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.
js8 3y ago
Interesting topic. I fondly remember, when I was doing geometry at school, I propagated a lot of errors.
PaulHoule 3y ago
What an answer to the problem of trisecting the angle! If you figure your angular accuracy is 0.3 degree or something you can construct a binary fraction like 341/1024.
- madcaptenor 3y ago
  Also trisecting a line isn’t far off from trisecting an angle, so constructing a pair of angles like 5/16 and 3/8 (=6/16) and then trisecting the chord between them would work. Wolfram MathWorld says that "[a]n approximate trisection is described by Steinhaus" (https://mathworld.wolfram.com/AngleTrisection.html https://mathworld.wolfram.com/AngleTrisection.html) and this is basically what I have in mind.
ogogmad 3y ago
You can do a mid-point construction using just a straight edge and some means of drawing parallels. Good example of projective geometry. ;)
nyc111 3y ago
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).
- wolfram74 3y ago
  The precision in the physical construction of the proof though does help in assessing it's validity. If the insight was completely divorced from the visual representation then you may as well be doing straight up symbolic algebra.

88 points by downboots 3y ago

throwaway81523 3y ago

2018

dang 3y ago
Added. Thanks!

deleted 3y ago

[deleted]

Sniffnoy 3y ago

Does anyone know where some of these follow-up posts might be?

bmacho 3y ago
> Unfortunately, I seem to have lost track of the references I had in mind. If anyone else can post suitable references, I’d be grateful. (2022)

konstantinua00 3y ago

[flagged]

justforasingle 3y ago

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.

gilleain 3y ago
Yes, I've learned some techniques after reproducing a lot of Islamic patterns, which can be done entirely with straightedge and compass (although frankly I often 'cheat' by using measures, grid paper and tracing paper, lol). Indeed the movement of the pencil can matter a lot, as can how good your compasses are. I've found that a bevelled ruler can be better if turned over so that the pencil rests against the bevel. Otherwise you have to take account of the width of the pencil lead and adjust the ruler position to take account of it. Certainly small errors in line position, length and (especially) angle can have a huge effect over a large diagram that has a lot of symmetries. It's surprising how noticeable even a small error of 0.5 mm between two lines is that should be colinear.
mbivert 3y ago
For that kind of things, lead-holders are preferable: you at least get a consistent lead width, despite the wearing, assuming you maintain the orientation of the pencil. An alternative is to sharpen your pencils with a utility knife & sandpaper: it's more versatile and less wasteful than a regular sharpener. For example, you could sharpen it to an edge (like a slotted screwdriver), and turn the pencil once in a while, so that the wearing helps preserving the edge. I've also dabbled with Islamic patterns recently. Side-note: I've found some high-quality, cheap second-hand compass sets, that used to be used for wood work a few decades ago (not made-in-China).

bee_rider 3y ago

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? 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. 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. Is that right? I somehow never managed to take a geometry class.

deleted 3y ago
[deleted]
Warwolt 3y ago
I think the circles are for when you create the arc using the compass. Presumably you already constructed points A and B and joined them into a segment, so there's now a point B to miss.
- bee_rider 3y ago
  If you are just trying to show the relationship between the lines, I don’t think there’s any reason to set A and B beforehand. Of course, it could be the case that there’s a multi-step proof where you do care about the positions of A and B in relation to some other things. But I think that would require doing the analysis with that in mind. Maybe the order in which you construct the more complicated proof can also benefit from picking certain points to be exact.
simplicio 3y ago
I think the construction is usually phrased: "given a line, to..", so it assumes a pre-existing line segment, rather then that being part of the construction itself.

bsder 3y ago

Welcome to laying out sheet metal: https://www.youtube.com/watch?v=9DypIDbAVTc https://www.youtube.com/watch?v=9DypIDbAVTc

adalacelove 3y ago

It's not a bug, it's a feature. All students know that any line goes through two big enough points. I was expecting something related to "exact predicates", like for example: Efficient Exact Geometric Predicates for Delaunay Triangulations 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)

auggierose 3y ago

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: https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIngyROBsUqcf_z https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIn... I am watching them while doing the dishes :-)

Etherlord87 3y ago

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). video: https://streamable.com/0xba64 https://streamable.com/0xba64 .gif: https://i.imgur.com/4Y33nrf.gif https://i.imgur.com/4Y33nrf.gif .blend file: https://e.pcloud.link/publink/show?code=XZvFUnZKrqbyl9h8ab3UCrJxdlBLHaW6qsX https://e.pcloud.link/publink/show?code=XZvFUnZKrqbyl9h8ab3U... 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...

deleted 3y ago
[deleted]

tromp 3y ago

> In this sense, the Apollonius perpendicular bisector construction appears to be sensitive to the errors of compass placement. 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.

js8 3y ago

Interesting topic. I fondly remember, when I was doing geometry at school, I propagated a lot of errors.

PaulHoule 3y ago

What an answer to the problem of trisecting the angle! If you figure your angular accuracy is 0.3 degree or something you can construct a binary fraction like 341/1024.

madcaptenor 3y ago
Also trisecting a line isn’t far off from trisecting an angle, so constructing a pair of angles like 5/16 and 3/8 (=6/16) and then trisecting the chord between them would work. Wolfram MathWorld says that "[a]n approximate trisection is described by Steinhaus" (https://mathworld.wolfram.com/AngleTrisection.html https://mathworld.wolfram.com/AngleTrisection.html) and this is basically what I have in mind.

ogogmad 3y ago

You can do a mid-point construction using just a straight edge and some means of drawing parallels. Good example of projective geometry. ;)

nyc111 3y ago

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).

wolfram74 3y ago
The precision in the physical construction of the proof though does help in assessing it's validity. If the insight was completely divorced from the visual representation then you may as well be doing straight up symbolic algebra.