n_m-clusters

General information about n_m-clusters

An n₂-cluster is n > 1 lattice points in R² such that no 3 are co-linear and no 4 are co-circular and all mutual distances between points are integers > 0.

In other words, a 2-dimension n₂-cluster is a collection of n lattice (grid points with integer (x,y) coordinates) on a flat plane such that no 3 lie on a straight line and no 4 lie on a circle and all of the distances between each pair of points are whole numbers > 0.

One need not be restricted to the R². We define n_m-clusters in R^m as follows:

• m and n are integers > 1
• n lattice points in R^m
• for all integer 0 < k < m, no k+2 points lie in k-dimensional plane (a k-dimensional affine subspace of R^m)
• for all integer 0 < l < m, no l+3 points lie on the surface of an l-dimensional sphere
• all mutual distances between the n lattice points are non-zero integers

n_m-cluster related terminology:

• prime n_m-cluster: an n_m-cluster where the greatest common divisor (gcd) of the mutual distances = 1
• primitive n_m-cluster: A prime n_m-cluster that has been rotated, reflected and translated into canonical form
  (Note to web page editor: define the canonical form here)

• equivalent n_m-cluster: two n_m-clusters are equivalent if and only if they can be made identical under the combined operations of rotation, reflection and translation
• nonequivalent n_m-cluster: two n_m-clusters that are not equivalent
  (all primitive n_m-clusters are both prime and nonequivalent)

• n-cluster: shorthand for n₂-cluster
• n(m)-cluster: alternate notation for n_m-cluster used in places where sub-scripting is not available or desirable

n₂-clusters

• n > 1 lattice points (points with integer (x,y) coordinates) in R²
• no 3 points lie on a straight line
• no 4 points lie on a circle
• all mutual distances between points are integers > 0

(0,0) (546,272) (132,720) (960,720) (546,-1120) (1155,-540)

The following are the smallest n₂-clusters:

(0,0) (1,0)
(0,0) (3,0) (0,4)
(0,0) (3,4) (3,-4) (6,0)
(0,0) (16,30) (-16,30) (0,-33) (56,0)
(0,0) (546,272) (132,720) (960,720) (546,-1120) (1155,-540)

A picture of the smallest 6₂-cluster is available.

n₃-clusters

• n lattice points (points with integer (x,y,z) coordinates) in R³
• no 3 points lie on a straight line
• no 4 points lie on a flat plane
• no 4 points lie on a circle

  NOTE: If no 4 points lie on a flat plane, then no 4 points can lie on a circle and therefore the above test may be skipped.

• no 5 points lie on the surface of a sphere
• all mutual distances between points are integers > 0.

(0,0,0) (54,28,12) (30,-20,60) (12,-56,-72) (-3,-76,-12) (-30,-84,12) (-72,0,96)
(0,0,0) (21,20,0) (-87,-4,-72) (108,24,-72) (126,-120,0) (-3,-172,96) (24,192,96)

The following are the smallest n₃-clusters:

(0,0,0) (1,0,0)
(0,0,0) (2,2,1) (2,2,-1)
(0,0,0) (2,2,1) (2,2,-1) (0,4,0)
(0,0,0) (6,6,3) (6,-6,3) (8,0,0) (8,0,6)
(0,0,0) (12,12,6) (-8,0,15) (-4,12,18) (20,0,15) (8,24,12)
(0,0,0) (54,28,12) (30,-20,60) (12,-56,-72) (-3,-76,-12) (-30,-84,12) (-72,0,96)

n₄-clusters

A n₄-cluster is:

• n lattice points (points with integer (x,y,z,w) coordinates) in R⁴
• no 3 points lie on a straight line
• no 4 points lie on a plane
• no 5 points lie on a hyper-plane (an R⁴ plane)
• no 4 points lie on a circle

  NOTE: If no 4 points lie on a flat plane, then no 4 points can lie on a circle and therefore the above test may be skipped.

• no 5 points lie on the surface of a sphere

  NOTE: If no 5 points lie on a hyper-plane (an R⁴ plane), then no 5 points lie on the surface of a sphere and therefore the above test may be skipped.

• no 6 points lie on the surface of a hyper-sphere (an R⁴ sphere)
• all mutual distances between points are integers > 0.

Randall and I (well Randall did all of the coding and I kibitzed and theorized on the side :-)) found these 7₄-clusters on 8 June 2001:

(0,0,0,0) (8,4,1,0) (0,8,-4,-8) (8,-4,5,-8) (-4,-4,11,4) (4,0,-6,-12) (12,8,-10,4)
(0,0,0,0) (8,8,4,0) (8,-8,4,0) (0,12,6,-4) (0,0,-9,12) (12,0,9,8) (8,4,10,12)
(0,0,0,0) (4,4,4,1) (4,-4,4,1) (-8,-4,4,10) (0,0,0,12) (0,12,-4,6) (0,12,12,6)
(0,0,0,0) (5,5,5,5) (8,-8,8,8) (-3,9,3,1) (5,1,11,-7) (0,-6,12,4) (0,-6,12,-12)
(0,0,0,0) (4,4,4,1) (-4,4,4,1) (-4,-8,4,10) (0,0,0,12) (12,0,-4,6) (12,0,12,6)
(0,0,0,0) (10,8,2,1) (10,-8,2,1) (0,0,0,12) (12,0,4,-6) (-6,-12,10,9) (12,0,-12,6)

and this 8₄-clusters on 10 June 2001:

(0,0,0,0) (8,8,8,8) (8,-10,4,-4) (0,-6,-12,4) (12,-3,-6,-6) (0,-9,-6,-18) (-12,-15,18,-6) (8,-25,-22,14)

The following are the smallest n₄-clusters:

(0,0,0,0) (1,0,0,0)
(0,0,0,0) (1,1,1,1) (1,1,1,-1)
(0,0,0,0) (1,1,1,1) (1,1,-1,1) (0,0,0,2)
(0,0,0,0) (4,3,0,0) (-4,3,0,0) (0,2,4,-4) (4,1,4,4)
(0,0,0,0) (4,2,2,1) (-4,2,2,1) (0,0,0,6) (0,6,2,3) (0,6,-6,3)
(0,0,0,0) (8,4,1,0) (0,8,-4,-8) (8,-4,5,-8) (-4,-4,11,4) (4,0,-6,-12) (12,8,-10,4)
(0,0,0,0) (8,8,8,8) (8,-10,4,-4) (0,-6,-12,4) (12,-3,-6,-6) (0,-9,-6,-18) (-12,-15,18,-6) (8,-25,-22,14)

n₅-clusters

A n₅-cluster is:

• n lattice points (points with integer (x,y,z,w,v) coordinates) in R⁵
• no 3 points lie on a straight line
• no 4 points lie on a plane
• no 5 points lie on a hyper-plane (an R⁴ plane)
• no 6 points lie on a 5-flat (an R⁵ plane)
• no 4 points lie on a circle

  NOTE: If no 4 points lie on a flat plane, then no 4 points can lie on a circle and therefore the above test may be skipped.

• no 5 points lie on the surface of a sphere

  NOTE: If no 5 points lie on a hyper-plane (an R⁴ plane), then no 5 points lie on the surface of a sphere and therefore the above test may be skipped.

• no 6 points lie on the surface of a hyper-sphere (an R⁴ sphere)

  NOTE: If no 6 points lie on a 5-flat (an R⁵ plane), then no 6 points lie on the surface of a hyper-sphere and therefore the above test may be skipped.

• no 7 points lie on the surface of a 5-sphere (an R⁵ sphere)
• all mutual distances between points are integers > 0.

Randall and I (as before) co-discovered these 8₅-clusters:

(0,0,0,0,0) (11,10,2,0,0) (13,6,2,0,4) (0,8,8,8,8) (4,12,8,4,4) (11,10,2,0,8) (13,6,2,8,4) (8,6,6,2,2)
(0,0,0,0,0) (6,6,6,6,0) (4,4,8,8,3) (10,2,10,6,4) (8,4,12,4,4) (4,8,8,12,1) (4,4,16,8,3) (8,4,12,8,1)
(0,0,0,0,0) (5,4,2,2,0) (0,8,4,8,0) (0,10,2,6,2) (5,14,0,8,2) (1,6,12,12,6) (8,10,2,14,6) (1,8,2,6,4)
(0,0,0,0,0) (5,4,2,2,0) (0,8,4,8,0) (0,10,2,6,2) (5,10,8,8,6) (1,12,6,6,12) (8,10,2,14,6) (1,6,4,8,2)
(0,0,0,0,0) (11,10,2,0,0) (13,6,2,0,4) (0,8,8,8,8) (4,12,8,4,4) (11,10,2,0,8) (9,14,2,8,4) (8,6,6,2,2)
(0,0,0,0,0) (8,4,4,4,3) (8,0,0,8,4) (6,10,2,6,7) (16,0,0,0,0) (10,2,10,6,7) (8,4,16,4,3) (8,4,4,8,6)
(0,0,0,0,0) (6,6,6,6,0) (0,12,4,8,1) (10,10,2,6,4) (8,12,4,8,1) (8,12,0,12,3) (4,16,4,8,3) (4,8,4,8,3)
(0,0,0,0,0) (6,6,6,6,0) (4,4,8,8,3) (10,2,10,6,4) (4,8,8,12,1) (8,0,12,12,3) (4,4,16,8,3) (8,4,12,8,1)
(0,0,0,0,0) (12,8,4,4,4) (12,5,10,2,4) (12,10,4,0,8) (12,6,12,0,0) (16,7,2,6,4) (4,13,10,10,12) (8,7,2,6,4)
(0,0,0,0,0) (8,0,0,0,0) (4,12,4,4,8) (8,10,10,4,3) (4,10,2,0,13) (12,8,4,0,10) (4,4,12,12,16) (4,8,4,8,6)

as well as these 9₅-clusters on 26 July 2001:

(0,0,0,0,0) (2,0,-2,4,-5) (4,-4,-4,4,0) (4,4,-8,0,-2) (-4,-4,0,8,-2) (-8,8,-4,4,6) (-4,4,-4,12,8) (-8,-8,-8,0,-8) (0,2,-4,6,-5)
(0,0,0,0,0) (0,8,0,0,0) (4,4,0,4,-4) (0,6,-3,6,0) (0,6,5,-2,-4) (8,4,-10,0,4) (8,4,-6,-8,-4) (0,-4,-10,-8,-4) (-8,4,-6,-8,-4)

The following are the smallest n₅-clusters:

(0,0,0,0,0) (1,0,0,0,0)
(0,0,0,0,0) (1,1,1,1,0) (0,0,0,2,0)
(0,0,0,0,0) (2,2,1,0,0) (0,0,1,2,2) (2,2,0,2,2)
(0,0,0,0,0) (2,2,1,0,0) (0,0,1,2,2) (2,2,0,2,2) (0,0,0,0,4)
(0,0,0,0,0) (2,2,1,0,0) (0,0,1,2,2) (2,2,0,2,2) (0,0,0,0,4) (0,4,1,2,2)
(0,0,0,0,0) (5,4,2,2,0) (5,2,0,4,2) (0,4,4,4,4) (0,0,0,8,0) (5,6,0,4,2) (6,4,4,4,4)
(0,0,0,0,0) (11,10,2,0,0) (13,6,2,0,4) (0,8,8,8,8) (4,12,8,4,4) (11,10,2,0,8) (13,6,2,8,4) (8,6,6,2,2)
(0,0,0,0,0) (2,0,-2,4,-5) (4,-4,-4,4,0) (4,4,-8,0,-2) (-4,-4,0,8,-2) (-8,8,-4,4,6) (-4,4,-4,12,8) (-8,-8,-8,0,-8) (0,2,-4,6,-5)

open n_m-cluster questions, conjectures & observations

n_m-cluster conjectures:

• Erdös/Noll infinite-or-bust n_m-cluster conjecture:
  For any m > 1, n > 2, there exists either 0 or an infinite number of primitive n_m-clusters.
• Noll infinite n_m-cluster conjecture:
  For any m > 1, n > 2, there exists an infinite number of primitive n_m-clusters.
• Noll/Rathbun computation observation:
  For any m > 1:
  • One will be able to find many (m+3)_m-clusters;
  • With bit more effort, one will also find some: (m+4)_m-clusters;
  • However, finding an (m+5)_m-cluster will be a significant computational challenge.

Open questions about n_m-clusters:

• No 7₂-cluster has ever been found. Do 7₂-clusters exist?
• No 8₃-cluster has ever been found. Do 8₃-clusters exist?
• No 9₄-cluster has ever been found. Do 9₄-clusters exist?
• No 10₅-cluster has ever been found. Do 10₅-clusters exist?

For more information see:

• Unsolved Problems in Number Theory, problem D20.
• '' n-clusters for 1 < n < 7,'' Landon Curt Noll and David I Bell; Mathematics of Computation, Vol 53, Number 187, July 1989, Pages 439-444.