32
votes
Accepted
Can a fixed finite-length straightedge and finite-size compass still construct all constructible points in the plane?
A bounded-length straightedge can emulate an arbitrarily large straightedge (even without requiring any compass), so the rusty compass theorem is sufficient.
Note that, in particular, it suffices to ...
- 12.4k
20
votes
How can we determine the center of a circle using a straightedge?
I do not think that it is possible.
Consider a projective map $f$ which preserves the circle; maps the center $O$ to a point $P\ne O$; and maps some point $Q$ to $O$. The line $QO$ is mapped to the ...
- 109k
19
votes
Accepted
Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?
There are three answers. Throughout let $qcl(F)$ be the quadratic closure of a field $F$ inside $\mathbb{C}$.
Part 1: Yes there is a quadratically closed field strictly between $qcl(\mathbb{Q})$ and ...
- 18.7k
14
votes
How can we find n points on a plane so that as many pairs of points as possible have the same distance?
The number is tabulated at OEIS. It seems that it's only known up to $n=14$ (and some scattered larger values). Links are given there to some papers on the topic. Evidently, no one knows how to do it ...
- 39.9k
10
votes
Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?
There are a lot of computations in this answer, so I hope there is no mistake. Answer to the small last question: there is a quadratically closed field strictly between the quadratic closure of $\...
- 471
7
votes
Accepted
Algorithm to decide whether two constructible numbers are equal?
Although this can be done using the complicated algorithms for general algebraic numbers, there’s a much simpler recursive algorithm for constructible numbers that I implemented in the Haskell ...
- 453
7
votes
How can we determine the center of a circle using a straightedge?
Edit: The following answer was written assuming that the ambiguous phrase “given a circle with diameter $AB$” in the question is to be interpreted as “given a circle having diameter $AB$” rather than “...
- 32.5k
6
votes
Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?
I've figured out how to do the finite graph case, in the negative. The ordering of quadratically closed fields is not dense in that case either. Throughout let $qcl(F)$ denote the quadratic closure ...
- 18.7k
5
votes
Accepted
How to shrink a square with minimal distortion?
Update. I'll show in the end of this post in the proof that the answer to the question is positive for all $\sigma_1\in (0,1)$.
The square is denoted by $\square$.
Set up. Let us fix $x,y\in (0,1)$ ...
- 28.9k
3
votes
Accepted
Domino tiling obtained from space-filling curves, is possible to predict basic properties?
I'm not sure I understood the question and answer given by the asker, but I gather that they are interested in decidability questions about some substitutive subshifts.
As far as I can tell, the ...
- 6,652
2
votes
What makes a geometric construction more or less stable?
In the case of regular polygons inscribed in a circle, you get addition of errors if you start with a chord that's supposed to be one side of the polygon and copy it all the way around in one ...
- 2,370
2
votes
Accepted
Reconstructing an ellipse from an arc, synthetically
Five points in general position lie on a unique conic. With the help of Pascal's theorem one can construct arbitrarily many points on the same conic.
EDIT. Answering the question about construction of ...
- 6,337
2
votes
Accepted
Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm
I can achieve $L(f - g) \leq (\frac{1}{2} + \frac{\pi}{4})\epsilon = (1.285\ldots)\epsilon$. Two reductions: (1) we can assume $|f(t)| < c$ for all $t \in (a,b)$ and (2) we can take $\epsilon = 1$.
...
- 42.8k
1
vote
How can we determine the center of a circle using a straightedge?
Please refer to Chapter 18 of "The Ruler in Geometrical Constructions" by Smogorzhevskii. There the author provides a provides a proof of sorts that in general where you have two(!) circles, ...
- 111
1
vote
Accepted
Geometric construction exercises
Yin Bon Ku created an assignment that solves my problem in GeoGebra.
It uses the Global JavaScript; see Event Listeners.
One has to create an activity with the boolean value ...
- 45k
1
vote
Accepted
Projections of particular simplex yielding boundary of a regular polygon?
I think you should be able to do this for arbitrary $m$ by taking the points $0$, $e_1$,
$$\cos\left(\frac{2\pi}{m}\right) e_1 + \sin\left(\frac{2\pi}{m}\right) e_2,$$
and
$$\cos\left(\frac{2\pi}{m}...
- 548
1
vote
Neusis constructions
baragar.faculty.unlv.edu/papers/TwiceNotch.pdf
Arthur Baragar proved the equivalence of neusis and conchoid-assisted constructions, and that all complex numbers constructible by neusis/conchoid lie on ...
- 111
Only top scored, non community-wiki answers of a minimum length are eligible