CITE THIS NOTEBOOK: Mathematical Games: Collection of Points and Lines by Ed Pegg. Wolfram Community FEB 17 2023.
This is part of live presentation series called Mathematical Games in which we explore a variety of games and puzzles using Wolfram Language. In this episode, we feature games and puzzles using points and lines.
Wolfram Demonstrations Project
Wolfram Demonstrations Project
Some of the Mathematics related demos I added on the Wolfram Demonstrations Project
Fixing A Messy Graph
Fixing A Messy Graph
In[]:=
messy=GraphData[{"RegularNonplanarDiameter",{3,3,16,11}}]
Out[]=
None of the embeddings look all that great. The last eleven embeddings are unit distance embeddings.
In[]:=
Graph[messy,VertexCoordinates->Thread[Range[16]->#]]&/@GraphData[{"RegularNonplanarDiameter",{3,3,16,11}},"AlternateVertexCoordinates"]
Out[]=
Throwing out a point gives a better structure:
In[]:=
Graph[Select[List@@@EdgeList[messy],Min[#]>1&],VertexLabels->"Name"]
Out[]=
Can we turn that into a nice unit-distance embedding? 1. Place the removed point at the origin. 2. CirclePoints[3] for a triangle of points. {2,3,4}3. Set a point at {0,-a} for three of the interior six points. {13,12,16}4. For a complete unit distance interior hexagon, the other point is at . {11,14,15}5. Use WFR CircleIntersection to get equations for two offset points within the outer nine. {5,8,9} and {7,6,10}6. Use WFR RealEuclideanDistance to get the distance between these two points. 7. Use NMinimize to solve for an exact unit distance.8. Use RootApproximant to find the exact solution.
0,a-
1
2
4-3
2
a
Out[]=
In[]:=
IsomorphicGraphQ[messy,nice]
Out[]=
True
How can we check that the exact solution of a geometric constraint problem is correct?
One weird way: check the discriminant.
One weird way: check the discriminant.
In[]:=
a=;b=RootReduce;NumberFieldDiscriminant/@{a,b}
-1+a(-1+a-2)-
2
a
9-3(-2+a)(-1+a)a(1+a)
4+4(-1+a)a
Out[]=
{25644883913205714777865715712,25644883913205714777865715712}
If they coincide, probably correct. If not, might not be the right root.
In[]:=
a=;b=RootReduce;NumberFieldDiscriminant/@{a,b}
-1+a(-1+a-2)-
2
a
9-3(-2+a)(-1+a)a(1+a)
4+4(-1+a)a
Find the number of vertices at distances 0, 1, 2, 3 for every vertex:
WFR GraphCoordinationSequence
WFR GraphCoordinationSequence
Lots of graphs (Dec 2017)
Lots of graphs (Dec 2017)
What’s the greatest number of unit lines you can get from vertices expressed by roots of 3 and 11?
A friend liked this demo a lot. “Have you considered expanding the roots from just 3 and 11 to 3, 5, 7 and 11?”
I replied: “Dear Aubrey, I don’t think that will help.”
A few months later, Aubrey de Grey found a 1581-vertex, non-4-colorable unit-distance graph.
That solved the Chromatic Number of the Plane. The biggest mathematical result of 2018.
Turns out that roots of 3, 5, 7 and 11 working together are very useful.
I was wrong.
I replied: “Dear Aubrey, I don’t think that will help.”
A few months later, Aubrey de Grey found a 1581-vertex, non-4-colorable unit-distance graph.
That solved the Chromatic Number of the Plane. The biggest mathematical result of 2018.
Turns out that roots of 3, 5, 7 and 11 working together are very useful.
I was wrong.
Two Points Define A Line
Two Points Define A Line
Three Points Define A Plane (WFR HessianPlane)
Three Points Define A Plane (WFR HessianPlane)
Three Points Define A Circle (Circumsphere, V10)
Three Points Define A Circle (Circumsphere, V10)
Circumcircle of triangle.
Four Points Define Two Parabolas (WFR FourPointParabolas)
Four Points Define Two Parabolas (WFR FourPointParabolas)
Newton proved it.
Order 4 projective plane with parabolas
Order 4 projective plane with parabolas
Any pair of points defines exactly one parabola.
Any pair of parabolas (and the unit circle) defines exactly one point.
Any pair of parabolas (and the unit circle) defines exactly one point.
Sylvester’s Four Point Problem
Sylvester’s Four Point Problem
Odds of all points on convex hull boundary?
Five Points Define A Conic Section (WFR FivePointConic)
Five Points Define A Conic Section (WFR FivePointConic)
Nine Points Define A Cubic (WFR NinePointCubic)
Nine Points Define A Cubic (WFR NinePointCubic)
If a line goes through two rational points, the third will be rational.
Magic Sums
Magic Sums
Pick three numbers from -7 to 7 that sum to 0, mod 15.
Are they on a straight line here?
We can add another point at infinity.
Are they on a straight line here?
We can add another point at infinity.
Orchard Problem Solutions
Orchard Problem Solutions
Solutions for 11, 16, 19 exceed the lower bound.
Do others exceed the lower bound? No-one knows!
Orchard Problem in 3D (WFR FindExtraordinaryLines)
Orchard Problem in 3D (WFR FindExtraordinaryLines)
149 points in 241 lines of 5.
What is the best way to collect lines of points in 3D space?
What is the best way to collect lines of points in 3D space?
Marden’s Theorem
Marden’s Theorem
Lucas-Gauss Theorem
Lucas-Gauss Theorem
Consider a complex polynomial.
The convex hull of the roots contains the roots of the derivative.
The convex hull of the roots contains the roots of the derivative.
Unit Distance Lines
Unit Distance Lines
6 Chromatic Graph in 3D
6 Chromatic Graph in 3D
“Two small 6-chromatic unit-distance graphs in R3” by Aubrey de Grey and Asger Haugstrup. GEOMBINATORICS Jan 2022
Rational 7
Rational 7
Seven points at integer distances.
Found in 2006 by Tobias Kreisel and Sascha Kurz
Found in 2006 by Tobias Kreisel and Sascha Kurz
Rational Distance Problem
Rational Distance Problem
Is there a point at rational distances from the vertices of a unit square?
A few thousand solutions with 3 rational distances.
No-one knows if 4 rational distances are possible!
A few thousand solutions with 3 rational distances.
No-one knows if 4 rational distances are possible!
Biggest Little Polyhedron
Biggest Little Polyhedron
27 Lines of a Cubic Surface
27 Lines of a Cubic Surface
The Clebsch surface.
A Plastic Pentagon
A Plastic Pentagon
19 points
Distances between them all powers of v = 1.1509639252577580357.
Is there a constant that works better? No-one knows!
Distances between them all powers of v = 1.1509639252577580357.
Is there a constant that works better? No-one knows!
Seven Touching Cylinders
Seven Touching Cylinders
Triangle Centers on a Sphere
Triangle Centers on a Sphere
Triangle Centers on a Poincare Disk
Triangle Centers on a Poincare Disk
Tetrahedron Centers (WTF Monge, Exspheres)
Tetrahedron Centers (WTF Monge, Exspheres)
Thomson Problem
Thomson Problem
Minimize charge of points on a sphere.
Unsolved over 12 points!
Unsolved over 12 points!
Heilbronn Problem
Heilbronn Problem
Middle Layers Danzer Graph
Middle Layers Danzer Graph
No Three In A Line
No Three In A Line
104 points on a 52×52 square. No 3 points are in a line.
Are larger solutions possible? No-one knows!
Are larger solutions possible? No-one knows!
No Repeated Distances
No Repeated Distances
Tetartoid
Tetartoid
Engel 38 Spacefiller
Engel 38 Spacefiller
Bracing a Heptagon
Bracing a Heptagon
Added to GraphData a long time ago. More recently, we found that this graph is rigid.
96_6 configuration
96_6 configuration
Every node has six lines. Every line has six nodes.
L. W. Berman, “Geometric Constructions for Symmetric 6-Configurations,” Rigidity and Symmetry, Springer, 2014, p. 83.
L. W. Berman, “Geometric Constructions for Symmetric 6-Configurations,” Rigidity and Symmetry, Springer, 2014, p. 83.
A Barycentric Self-dual 4-Configuration
A Barycentric Self-dual 4-Configuration
Every line is a point. Every point is a line.
Nautilus with Fractal triangle
Nautilus with Fractal triangle