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I don't understand what a right angle is.

Of course, I know what a right angle is, but I feel I don't understand it.

I'm looking at Euclidean geometry of the plane.

When looking at it from analytic geometry, everything is fine, but there the concepts of orthogonality, distance and dot product give a right angle for free.

From synthetic geometry everything gets fishy.

Euclid described a right angle as the angel where two intersecting lines produce 4 equal angles.

But how would he decide if the angles are equal without silently assuming that an angle can be rotated without changing.

From what I have seen and played with, it looks like a right angle cannot be defined without the concept of rotation and/or length, but I'm totally new to synthetic geometry and possibly miss some fundamental facts.

Edit (to make the question more specific):

Is there an incident structure that is compatible with the analytical definition of the Euclidean plane, based on points and lines that defines a right angle?

Edit (background clarification):

The starting point behind the question is computer geometry:

floating point calculations are messy (unavoidable numerical errors)

square roots are messy (numerical errors and performance issues)

trigonometric functions are messy (numerical errors and severe performance issues)

The primary root of the question is: do we need square roots and angles at all, and where can we avoid them?

Gyro Gearloose
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  • Many angels in your question... Its Easter... – Jean Marie Apr 13 '20 at 12:21
  • This said, it's a real question at the origin for example of non-euclidean geometries... – Jean Marie Apr 13 '20 at 12:23
  • That "rigid motions" preserve length, angle, area is implicit in Euclidean geometry. – Gerry Myerson Apr 13 '20 at 12:39
  • I would assume it should be sufficient to postulate that such an intersection (with equal angles) exists. – user Apr 13 '20 at 12:45
  • @GerryMyerson that implies that the concept of length, angle and area is known and strictly defined ,,, not the best start to define length, angle and area – Gyro Gearloose Apr 14 '20 at 10:24
  • Two line segments have the same length if there is a rigid motion that carries one to the other. Two angles have the same measure if there is a rigid motion that carries one to the other. Euclid may never have explicitly said so, but this is what he had in mind. – Gerry Myerson Apr 14 '20 at 12:34
  • @GerryMyerson I think this is a circular definition: "same length" of line segments by rigid motion and rigid motion by preserving line length. – Gyro Gearloose Apr 15 '20 at 15:20
  • Rigid motions are translations, rotations, and reflections. Euclidean geometry is the study of quantities invariant under rigid motions. One of these quantities gets the name "length", another gets the name "angle", another gets the name "area", and so on. – Gerry Myerson Apr 15 '20 at 22:37
  • @GerryMyerson You are perfectly right iff you start with those quantities as primitives, underived things. – Gyro Gearloose Apr 16 '20 at 12:16
  • Well, you have to start somewhere. – Gerry Myerson Apr 16 '20 at 12:18
  • @GerryMyerson The starting point behind the question is computer geometry: floating point calculations are messy (unavoidable numerical errors), square roots are messy (numerical errors and performance issues) and trigonometric functions are messy (numerical errors and severe performance issues). The primary root of the question is: do we need square roots and angles at all, and where can we avoid them? – Gyro Gearloose Apr 16 '20 at 12:41
  • Well, why didn't you say so? I would never have guessed any of that from the way you presented the problem. Why not put that in the body, so readers will know what's really going on? Anyway, are you familiar with Norm Wildberger's Rational Trigonometry, where he uses "quadrance" and "spread" instead of length and angle? https://en.wikipedia.org/wiki/Rational_trigonometry – Gerry Myerson Apr 16 '20 at 12:47
  • I say, are you familiar with Wildberger's work? – Gerry Myerson Apr 18 '20 at 03:46
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    @GerryMyerson Thanks for the link to rational geometry, I've never heard of Wildenberger before. Looks quite a lot like what I was searching for. – Gyro Gearloose Apr 20 '20 at 08:23
  • @GerryMyerson from Wildenberger's work (p41): "To build up mathematics properly, axioms are not necessary." That's where I stopped reading. – Gyro Gearloose Apr 20 '20 at 15:26

1 Answers1

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You are right to be puzzled, and you're in good company.

Euclid's Definition 10 is

When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Postulate 4 asserts

That all right angles equal one another.

(https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html)

Implicit in his notion of the equality of geometric figures is the idea that one can be translated and rotated to coincide with the other.

In order to connect Euclid's notion of right angles to the one you know from coordinate geometry you have to introduce a coordinate system. To do that you need the notorious fifth postulate. \

Related: Are the proofs for the properties of parallel lines, and that a triangle has 180 degrees, inherently tautological?

Is Pythagoras' Theorem a theorem?

Ethan Bolker
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  • The fifth postulate implies not only the existence of angles but also that they can be added and compared with "less than". – Gyro Gearloose Apr 14 '20 at 10:25