Questions tagged [geometric-constructions]

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How can we determine the center of a circle using a straightedge?

Given a circle with diameter AB, how can we determine the center of the circle with a straightedge (we cannot measure lengths, cannot measure angles, or draw parallel lines,... We can only draw ...
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31 votes
3 answers
2k views

Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?

Consider the hierarchy of relative geometric constructibility by straightedge and compass. Namely, given a geometric figure $B$, a set of points in the plane, we define that geometric figure $A$ is ...
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3 votes
0 answers
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Geometric construction exercises

Many of you know dynamic geometry exercises in Euclidea; if not, here is one example. It lets you do a geometric construction and sends a message once you achieve the result. I am looking for a way to ...
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2 votes
0 answers
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The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?

I asked this question over a year ago on Math.StackExchange but I didn't get an answer. In his famous treatise On spirals, Archimedean used a spiral to square the circle and trisect an angle. There ...
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Compass and straightedge construction of Poncelet polygons

Gauss–Wantzel theorem states that A regular n-gon is constructible with straightedge and compass if and only if $n = 2^kp_1p_2...p_t$, where $p_i$'s are distinct Fermat primes (A Fermat prime is a ...
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10 votes
1 answer
661 views

How to shrink a square with minimal distortion?

$\newcommand{\CO}{\text{CO}_2}$ $\newcommand{\euc}{\mathfrak{e}}$ $\newcommand{\SO}{\text{SO}_2}$ $\newcommand{\al}{\alpha}$ $\newcommand{\dist}{\text{dist}}$ $\newcommand{\Lip}{\text{Lip}_{\text{inj}}...
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5 votes
0 answers
194 views

Are there any neusis-hard/neusis-complete problems?

I have lately been enjoying Richeson's Tales of Impossibility (see MAA review), an accessible book on the famous problems of Euclidean geometry including angle trisection/cube doubling/heptagon ...
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28 votes
2 answers
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Can a fixed finite-length straightedge and finite-size compass still construct all constructible points in the plane?

I am hoping that the brilliant MathOverflow geometers can help me out. Question 1. Suppose that I have a fixed finite-length straightedge and fixed finite-size compass. Can I still construct all ...
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4 votes
2 answers
418 views

Domino tiling obtained from space-filling curves, is possible to predict basic properties?

Periodic and aperiodic domino tiling systems can be obtained by the following construction rules: Draw a regular square grid n×n of n2 cells. Select a space-filling curve that is consistent with ...
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10 votes
1 answer
964 views

How can we find n points on a plane so that as many pairs of points as possible have the same distance?

There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
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Examples of geometrical interpretations for sequences of particular values of Dirichlet series

The remark [1] (in Spanish) shows a geometric interpretation (linking two sequences) of particular values of a given Dirichlet series, that are $\zeta(k)$ and $\zeta(2k)$. I wondered about if it is ...
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1 answer
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Projections of particular simplex yielding boundary of a regular polygon?

What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?
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6 votes
1 answer
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Algorithm to decide whether two constructible numbers are equal?

The set of constructible numbers https://en.wikipedia.org/wiki/Constructible_number is the smallest field extension of $\mathbb{Q}$ that is closed under square root and complex conjugation. I am ...
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2 votes
0 answers
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What makes a geometric construction more or less stable?

I'm not entirely sure if this is "research level" math or not, but I asked on Math.SE and it was suggested I try asking here, so hopefully it's of interest to this community. (Original question on M....
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36 votes
1 answer
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Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid)

Let me start with the context. This is definitely not a "research level" question, but I'm hoping that the research community will be able to settle for me whether or not a particular construction ...
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3 votes
1 answer
291 views

Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses. Also it is possible to construct straight line ...
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5 votes
1 answer
162 views

Antoine's Necklace and positive Hausdorff/Lebesgue measure

I have the following question: The usual construction of the Antoine's Necklace produces a Cantor of $1$-dimensional Hausdorff measure in $\mathbb R^3$. I would like to know whether one could adapt ...
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4 votes
1 answer
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Focus of parabola using only a ruler

It is an easy exercise that using ruler and compass one find the focus of a given parabola. Can one do the same using only a ruler? -- if not, why?
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9 votes
0 answers
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How can we join two points with a small ruler? [closed]

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this? Imagine a method that works when $AB$ is really huge and ...
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  • 892
32 votes
2 answers
967 views

Term for "uncheckable constructions"

Is there a term for "uncheckable geometric constructions"? Say, Angle Trisection and Doubling the Cube are checkable; i.e., if the answer is given one can do finite Compass-and-straightedge ...
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10 votes
0 answers
647 views

The construction of the 257gon

If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...
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2 votes
3 answers
2k views

Mid point with set square?

Is it possible to construct the midpoint of a segment in the hyperbolic plane using the set square only? With the set square one can draw the line through the given two points and drop the ...
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1 vote
1 answer
243 views

Algebraic characterization of points constructible by compass and straightedge

The typical characterization of points constructible by compass and straightedge is the following: Let $S\subseteq\mathbb{C}$ with $0,1\in S$, $K_0 = \mathbb{Q}(S\cup \bar{S})$ and $a\in\mathbb{C}$. ...
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  • 253
5 votes
2 answers
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How to draw Archimedean-Galileo spiral?

It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and ...
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5 votes
4 answers
994 views

What are the lengths that can be constructed with straightedge but without compass?

Most field theory textbooks will describe the field of constructible numbers, i.e. complex numbers corresponding to points in the Euclidean plane that can be constructed via straightedge and compass. ...
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6 votes
2 answers
630 views

Bolyai's construction

Here is Bolyai's construction (in Klein model), which I learned recently from answer of Will Jagy to this question. It is a Compass-and-straightedge construction of asymptotically parallel line in ...
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9 votes
2 answers
1k views

Is there a ruler and compass construction of the common perpendicular of two geodesics in H^3?

Assume we have two geodesics in the Poincaré ball model of $\mathbb{H}^3$, viewed as arcs intersecting the boundary of and contained in the Euclidean unit sphere in $\mathbb{R}^3$. Is there a ruler ...
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9 votes
2 answers
2k views

Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?

Is there a compass and straightedge construction of parallel lines in hyperbolic geometry? That is, given a line and a point not on the line, construct a line parallel to the given line.
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27 votes
6 answers
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How fast are a ruler and compass?

This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO. Consider the standard assumptions ...
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14 votes
3 answers
2k views

Neusis constructions

Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis? See http://en.wikipedia.org/wiki/Constructible_number and http://en.wikipedia....
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16 votes
1 answer
1k views

Origami Constructions: Intersecting two Circles

It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see: R. Geretschlager. Euclidean Constructions and the Geometry ...
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7 votes
3 answers
3k views

how to construct a spherical dodecahedron?

using only a spherical ruler (to construct great lines) and a pair of compasses, how can you construct a regular dodecahedron on the surface of the sphere? thank you very much.
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3 votes
5 answers
1k views

Approximate solutions for trisecting the angle or squaring the circle

Hello all, it is well-known by transcendence results or Galois theory that famous geometric problems such as trisecting an angle or "squaring the circle" (i.e. given a disk of radius 1 construct a ...
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  • 3,485
4 votes
1 answer
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On using field extensions to prove the impossiblity of a straightedge and compass construction

Let $z \in \mathbb{C}$. Consider the following statements: The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$. There is a field extension $K / \mathbb{...
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