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How do you know a line is straight? How can you check in a practical way if something is straight - without assuming that you have a ruler? How do you detect that something is not straight?

If you fold a piece of paper the crease will be straight — the edges of the paper needn’t even be straight. This utilizes mirror symmetry to produce the straight line.

Carpenters also use symmetry to determine straightness — they put two boards face to face, plane the edges until they look straight, and then turn one board over so the planed edges are touching.They then hold the boards up to the light. If light passes between the boards the edges are not straight.

Are there other ways to determine if a line is straight?

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    $\begingroup$ You might be interested in the book "How Round Is Your Circle?". It discusses this exact problem, among others. $\endgroup$ – Michael Biro Sep 27 '17 at 1:16
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    $\begingroup$ @MichaelBiro thank you so much for that suggestion. I have been looking for some books that can explain this to me. I will definitely look into that. $\endgroup$ – Hidaw Sep 27 '17 at 1:18
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    $\begingroup$ When I fold a piece of paper the crease is always curved and when I try to straighten it out it gets crumples. I try to judge straightness (say of a line on a graph) by viewing it as edge on as I can, with my eye close to the corner of the page. $\endgroup$ – kimchi lover Sep 27 '17 at 1:18
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    $\begingroup$ This is a deep question, but it is physical, because straightness of a real line is defined by way which we measure distances. Light rays are straight because they always go the shortest ways, along so-called geodesics. $\endgroup$ – Alex Ravsky Oct 1 '17 at 9:14
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    $\begingroup$ ? Perhaps you mean How can a computer "see" straight lines : Finding Edges and Straight Lines . $\endgroup$ – Han de Bruijn Oct 4 '17 at 12:57
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If you pull a string tight, it will approximate a straight line. Strings aren't too hard to come by in the natural world.

This is using the fact shortest-length paths are geodesics, related to the symmetry of flat $\mathbb R^3$.

Linkages are useful for solving equations mechanically. The Peaucellier–Lipkin linkage uses the principle of inversion through a circle, an operation taking lines and circles to lines and circles. If I remember correctly, the linkage inverts over a circle a circle tangent to it to a line.

Perhaps you only care about collinear points. Light travels in approximately straight lines, so insert pushpins at the points and look down them to see if they line up. This uses the fact that perspective projection carries lines passing through the focus to a point.

Neither of these help with the carpentry example of a flat surface.

You could carefully grow a crystal and cleave it along a crystal plane, taking advantage of the crystal symmetry.

Other things to do with string:

Fixing one end of the string and moving the other, you get a circular arc.

Putting a loop of string around two thumbtacks pulling a triangle tight, the vertex traces out an elliptical arc.

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  • $\begingroup$ String will sag, unless you’re stretching it vertically. A method I use when doing woodworking is to sight along the edge. Deviations from straightness will show up pretty well under the perspective. $\endgroup$ – Lubin Sep 27 '17 at 2:04
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I don't know if this is what you want. But your question reminds me of an exercise in differential geometry.

A smooth curve $C$ in $\mathbb{R^3}$ is a straight line if and only if the curvature of $C$ is everywhere zero.

of course as Ravsky mentioned in comment, the concept of "straight" is hard to define in non-Euclid geometry. Instead, we use the concept of "geodesics" to generalize "straight line".

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One practical way to test the straightness of something in real life is to use a laser.

One way to do it geometrically is to construct two perpendicular lines (at two arbitrary points on your "line) if the perpendicular lines are perfectly parallel they will never intersect (no matter how you choose the two arbitrary but not identical points). If you find locations two points such that the perpendicular lines not perfectly parallel there will be a point of intersection at a finite distance.

Sorry for the ugly picture enter image description here

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