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Given the ease with which we can geometrically multiply and divide straight-line segments, i would like to ask for guidance on the same problems on the circle, not necessarily using "straight edge and compass", but any possible method:

Given two arcs of length x and y on the unit circle, construct arcs of length $x\cdot y, \frac{x}{y}, \,\,\frac{x}{n} \text{ for }n=2,3,4,\ldots$.

The only results i'm aware of are those of the theory of constructible polygons, with straight edge and compass.

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  • $\begingroup$ If you're not restricted to ruler and compass, why not jus multiply $x$ by $y$ to get a number $z$, and then use the arclength integral to determine a point that's distance $z$ from, say, $(1, 0)$? (This assumes that $z \le 2\pi$, or that "wrapping" is allowed). If you do restrict to ruler and compass, then computing $x/3$ is a problem, for this amounts to trisecting the angle. $\endgroup$ Commented Jul 8, 2019 at 11:42
  • $\begingroup$ @John Hughes: but how do we get the number z = x by y, if x and y are two points given on the circle? $\endgroup$
    – exp8j
    Commented Jul 8, 2019 at 11:48
  • $\begingroup$ You didn't ask about dividing or multiplying points, you asked about arcs. And to find out the arclengths, you...integrate the arclength integral. It's really, really unclear what you're asking here, or what could possibly be a satisfactory answer. $\endgroup$ Commented Jul 8, 2019 at 11:51
  • $\begingroup$ @John Hughes: i mean two points x and y, specifying arcs of length x and y, measured from a given initial point on the circle. $\endgroup$
    – exp8j
    Commented Jul 8, 2019 at 11:56
  • $\begingroup$ Can you explain what it means to geometrically multiply and divide line segments? $\endgroup$ Commented Jul 8, 2019 at 12:22

2 Answers 2

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Use a thick disk and a thread. You can roll/unroll and straighten the thread to convert from arc to line segment.

Alternatively, use an Archimedes' spiral. The polar angle and the modulus perform the same transformation.

enter image description here

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  • $\begingroup$ Is there a proof that x*y and x/y can't be done with straight-edge and compass? $\endgroup$
    – exp8j
    Commented Jul 8, 2019 at 15:19
  • $\begingroup$ @exp8j: read the comments in the OP. $\endgroup$
    – user65203
    Commented Jul 8, 2019 at 15:20
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OK. Let's assume a unit circle. Given points $X$ and $Y$ with arclengths $x\in \Bbb R$ and $y\in \Bbb R$ measured counterclockwise from the initial point $P = (1, 0)$, compute, using the axioms of the real numbers, the number $xy$. Then let $U = (\cos(xy), \sin(xy))$; the arclength counterclockwise from $P$ to $U$ is then $xy + 2k \pi$, for some integer $k$. The same approach works for all the other possibilities.

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  • $\begingroup$ sorry if i didn't make it clear: x and y are not given numbers, but points on the circle specifying two arcs relative to an origin. The problem is to define the product and the ratio geometrically (to define the sum is easy, we just put the two arcs side by side). $\endgroup$
    – exp8j
    Commented Jul 8, 2019 at 12:27
  • $\begingroup$ OK. So let $\theta_x = atan2(v, u)$, where $(u, v)$ are the coordinates of the point $x$, and similarly for $\theta_y$. As for "define geometrically," that's not precise enough to have any useful meaning. After all, we've defined all the operations on the reals geometrically (as you observed yourself),so now using real numbers is...geometric? Not geometric? I dunno. Anyhow, my real point was that until you refine your question to make it well-defined, it's not very interesting. Personally...I'm done. $\endgroup$ Commented Jul 8, 2019 at 13:02

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