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Questions tagged [geometric-construction]

Questions on constructing geometrical figures using a limited set of tools. The compass and straightedge are almost always allowed, while other tools like angle trisectors and marked rulers (neusis) may be allowed depending on context.

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982 views

Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?

In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
7
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180 views

Optimal Compass and Straightedge Constructions

I was recently looking over some Islamic geometry patterns, and was struck by the complexity of the constructions needed to create seeming simple patterns. This got me wondering regarding optimal ...
7
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0answers
141 views

What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
7
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307 views

Can one trisect $\arccos(6/7)$?

Is this proof correct? Proof: Here $\theta = \arccos(6/7)$. Now to show we can't trisect $\theta$, we show that $\theta/3$ is not constructible by finding the irreducible polynomial in $\mathbb Q[x]$,...
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0answers
71 views

Constraints on conical coffee cup constructions of cardioids & catacaustics

The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) ...
5
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0answers
299 views

Find the intersection of two lines entirely outside the given sheet of paper by straightedge alone

This is a problem from Courant:"Two straight lines entirely outside the given sheet of paper are each given by two pairs of straight lines intersecting at points of the lines outside the paper. ...
5
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0answers
80 views

Geometrical construction of 7th roots of unity given parabola x^2

Given parabola $x^2$ on plane, how can I construct 7th roots of unity? I was straying with my only idea, that sum of squares of real and imaginary part of roots of 1 equals 1 and belongs to $\mathbb{...
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0answers
164 views

Constructional proof of ellipse property

I came across the fact that the following function defines a family of ellipses with focal distance $f$, parameterized by the value of the function: $$\frac{x-f+\sqrt{(x-f)^2+y^2}}{x+f+\sqrt{(x+f)^2+...
4
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0answers
98 views

Minimal number of steps to construct $\cos(2 \pi /n)$

My question is related to this previous one. I was wondering what is the minimal number of steps $S(a)$ to construct a number $a \in \mathbb R$ that is constructible (as defined here). For instance, ...
4
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0answers
474 views

Dynamically generate Goldberg polyhedra G(m,n)

In these pages the author provided a lot of info about some Goldberg polyhedra (http://en.wikipedia.org/wiki/Goldberg_polyhedron): http://dmccooey.com/polyhedra/DualGeodesicIcosahedra.html http://...
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0answers
30 views

Relation between $[L \cap M : K \cap M]$ and $[L : K]$, and the Wantzel theorem

The well-known Gauß-Wantzel Theorem states that a real number $x$ can be constructed using straightedge and compass only if the minimal polynomial of $x$ (over the field $\mathbf Q$) has degree of ...
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0answers
78 views

Recognizing integer Pythagorean triples geometrically and whether two points constructed on a line are the same

This is suggested by the question Proof that the sum of the even side and the hypotenuse of a coprime (and positive) Pythagorean triple is a square number and the fact that I really like Euclid's ...
3
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0answers
47 views

Ruler-and-Compass metric: draw a line from any point to separate the area of a triangle?

Given a triangle $ABC$ and a point $X$, is it possible to only use Ruler and Compass, to draw a line $l$ through $X$, such that $l$ will split $ABC$ to two parts with equal area?
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0answers
1k views

Cube root of a line

Well this may be simple but I am not getting it. Give a line segment (of length $l$)(and a segment of unit length if you require) how to construct a line of length $l^{1/3}$ with only a straight ...
3
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0answers
97 views

Is this curve defined by an envelope construction known?

Consider the following construction. Start with the standard envelope construction of a cardioid: on a circle, join each point $\theta$ to $2\theta$. Only, instead of joining with a line, join with ...
3
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0answers
59 views

how do the conic sections add to the possibilities of geometric construction?

If we are limited by what we can construct with compass and straight edge, then what becomes possible by expanding our toolkit to include all conic sections? The tools would be based on already ...
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0answers
101 views

Algebraic number of degree four that cannot be constructed with ruler and compass

The real number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$, where \begin{equation*} a:=\sqrt[3]{\frac{9+\sqrt{65}}{4}}+\frac{1}{\sqrt[3]{\frac{9+\sqrt{65}}{4}}}=\frac{\sqrt[3]{18+2\cdot\sqrt{65}...
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504 views

History of the neusis construction of cube roots?

A simple neusis (marked ruler) construction of $\sqrt[3]{2}$ is given in many places, for example wikipedia. My question is: what is the history of this construction? As far as I can determine, all ...
3
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0answers
218 views

Exercise $7.7$ page $82$ Ian Stewart, Galois Theory

Prove that an angle $\theta$ can be trisected by ruler and compasses iff the polynomial $4t^{3}-3t-\cos\theta$ is reducible over $\mathbb{Q}\left(\cos\theta\right)$
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12k views

Division of a line segment in the given ration internally

Before I state my problem description, would like to describe problem which was stated before my problem. So it is like this Given a line segment $AB$. You are required to divide it internally ...
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54 views

Find the line that is closest to 4 skew lines

If I have 4 skew lines in $\mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them? I know that with 3 skew lines, there is always a line that intersects all of them, in fact ...
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0answers
113 views

Geometric construction of golden angle

Is it possible to construct the golden angle using only a compass, ruler and pencil?
2
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0answers
77 views

What is this “easy application of the Pythagorean theorem”?

I am reading Stillwell's Elements of Algebra, he gives this figure: $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ And then: What is this easy application of the Pythagorean theorem? I ...
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0answers
55 views

Given sufficiently large sample of flight paths, is it possible to derive the map?

Background Suppose we do not have a map of the world but we can fly from a random point towards a random direction. The speed are the same for all flights and remain constant throughout each flight....
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0answers
147 views

Relative difficulty of angle trisection and cube roots

It is well-known that arbitrary angles cannot be trisected by using a straightedge and a compass. The proof of this fact is done by reduction to unsolvability of cubic equations with square roots (see ...
2
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0answers
54 views

Question regarding constructibility of a point

Given 2 points $ A $ and $ B $ in the complex plane with $ AB=1 $, is it possible to construct with unruled straightedge and compass a point $ C $ on the line $ AB $ such that $ AC \cdot BC^{2}=1 $? ...
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0answers
104 views

Straightedge and compass theory in three dimensions

I'm looking for a reference on the theory of straightedge and compass constructions in three dimensions akin to Euclid's Elements in two dimensions. More specifically, I mean a theory of geometric ...
2
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0answers
100 views

A circle can include all but one of n points, but which one can it be?

The answers to the question "Circle enclosing all but one of n points" demonstrate that, given $n$ points, it is possible to construct a circle such that all but one of the points is inside the circle ...
2
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0answers
182 views

constructing an equilateral triangle in the Beltrami klein model

I am puzzeling with the following: Using the beltrami klein disk of hyperbolic geometry (see https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model ) (PS not the poincare disk model) and given ...
2
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0answers
103 views

Thread constructions in the Poincaré's disc

I just came across the following image (source) and realised something that should have been obvious to me a while ago: it should be possible to construct the envelopes of curves in the Poincaré disc ...
2
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0answers
289 views

Huzita Axiom 6 - Computing the Origami Trisection of an Angle

The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be ...
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0answers
236 views

Smallest field containing $\mathbb{Q}$ and closed under square root

I'm following Isaacs' Algebra and I need to prove that the field $K$ of constructible numbers is the smallest subfield of $\mathbb{C}$ such that is closed under taking square root. I already know ...
2
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0answers
78 views

Is this element constructible from this elements?

Let the figure below. According to same notation of the figure verify if it's possible to construct the point $\displaystyle \zeta=e^{\frac{2\pi i}{13}}$ with straight-edge and compass from $X=\{0,1,\...
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0answers
97 views

Computing an area with geometric methods

Consider the construction below (it is self-evident, no tricks or misleading drawing) What is the area enclosed by the curve that goes through the points H,J,M,L ? I solved the problem using ...
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0answers
321 views

The Basel Problem and Theodorus' Spiral

I've been trying to find a classical solution to the famous Basel Problem solved by Euler. To those unfamiliar the problem is to find the sum infinite series made up of the reciprocals of square ...
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0answers
43 views

Making an approximation for nth index using Prime number theorem

Abstract: Using of Gauss’s Prime number theorem for finding the number of primes can also tell us which primes are located for the desired index. But it will give a rough approximation because prime ...
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0answers
34 views

Finding the region surrounded by light emitted from the vertices of a triangle through construction

P, Q and R represent the positions of three radio beacons. P is 450 km from Q, Q is 475 km from R, and R is 300 km P. Signals from P have a range of 300 km, Q has a range of 350 km and R has a range ...
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0answers
77 views

Given $\overline{AB}=c$, $\overline{AC}=b$, $m_a$, construct the triangle $ABC$?

Given $\overline{AB}=c$, $\overline{AC}=b$, $m_a$, construct the triangle $ABC$. I did the following: And then used the compass to draw the distances $c,b$: I've drawn the paralelogram: ...
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0answers
133 views

Geometry: Division of a line segment in a given ratio

In geometric construction while dividing a line segment in a given ratio say $m:n$, I have come across the texts that a ray should be drawn making an acute angle with the given line segment. I'm not ...
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0answers
34 views

the Support function of reuleaux triangle

I am looking for the explicit support functions (and also reuleaux polygones ) but all I find is general properties , could you give a Book or a paper when can i found the expolicit support ...
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58 views

Given the angles and diagonals of a quadrilateral, construct the quadrilateral using only a straightedge, a pencil, and a compass

I tried to do this by constructing 1 angle, and then constructing 2 circles with their radii equal to the diagonals with their centres on the vertex of the angle, but I couldn't get any further from ...
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0answers
45 views

Constructibility of the $\sqrt[n+1]{2} + \sqrt[n+3]{2}$ where $n$ is even

Let $n$ be even. Are there any constructible numbers of the form $\sqrt[n+1]{2} + \sqrt[n+3]{2}$? Attempt My initial hunch is no. Because the $\sqrt[n+1]{2}$ is a root of the polynomial $f(x)=x^{n+1}...
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0answers
16 views

Reference for principles regarding construction of geometric figures with fewest steps as possible.

Is there such guiding principle? I have played geometry construction games, and most of the time I do too many steps making it hard to proceed.
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0answers
180 views

quintsect an angle with a compass and a straight edge

I require to construct a reasonably accurate $9^{\circ}$. And quickly. I learn that the quickest way is to make a straight line of $180^{\circ}$, bisect it (to make $90^{\circ}$), bisect that angle (...
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0answers
402 views

How to construct the circumcenter of a triangle using a compass ONLY.

I just figured out how to find the midpoint between two points using just a compass and no straight edge. A similar approach can be found in this question: Constructing the midpoint of a segment by ...
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0answers
114 views

Geometric / Intuitive construction of the rotation axis of a 3D rotation matrix?

I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix. To put the problem in more familiar terms, let's assume you have the ...
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0answers
87 views

Relation between sqrt and ratio in ruler and compass?

The construction in most vote reply in Compass-and-straightedge construction of the square root of a given line? uses similar traingles and uses $$\frac{AC}{AD}=\frac{AD}{AB}$$ to compute square root. ...
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0answers
167 views

Pyramid with unit sides inside a cube

Let $ABCDEFGH$ be a unit cube with base $ABCD$. Let $P$ be the top of the pyramid with base $ABCD$ and all edges of length $1$. One has a standard 2-dimensional projection of this cube on the back ...
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0answers
59 views

How do you visualize ridge, roof and step edges?

I am reading about Canny algorithm in the book Academic Press - Handbook Medical Imaging Processing Analysis where it is written that the algorithm was originally developed for antiasymmetric edges $(...
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0answers
91 views

Why $\pi$ is not Constructible with Circumference Length

If we use a compass to draw a circle with a diameter of length 1, then the circumference is $\pi$. From the definition given here (http://en.wikipedia.org/wiki/Constructible_number), it seems to me $\...