Clusters 1 7 2

General information about n_m-clusters

1. Clusters 1 7 2 Iso
2. Clusters 1 7 2 Mods
3. Clusters 1 7 20
4. Clusters 1 7 2 0

Figure 7-1 shows a histogram for the distribution of occupations in a cluster of customer data. In this cluster, about 13% of the customers are craftsmen; about 13% are executives, 2% are farmers, and so on. 2 Site 3 and Cluster 1-2 9.3 1 Cluster 1-2 and Cluster 4-5 10.0 29 Combine site 6 and cluster 4-5 Combine site 4 and cluster 1-2 Combine cluster 1-2-3 and cluster 4-5-6.

An n₂-cluster is n > 1 lattice points in R²such that no 3 are co-linear and no 4 are co-circular and all mutual distancesbetween points are integers > 0.

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

2 nodes, which will be used to create the cluster. In this example, the nodes used are z1.example.com and z2.example.com. Network switches for the private network, required for communication among the cluster nodes and other cluster hardware such as network power switches and Fibre Channel switches. A computer cluster is a set of loosely or tightly connected computers that work together so that, in many respects, they can be viewed as a single system. Unlike grid computers, computer clusters have each node set to perform the same task, controlled and scheduled by software.

One need not be restricted to the R².We define n_m-clusters inR^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: ann_m-cluster where thegreatest common divisor (gcd) of the mutual distances = 1
• primitive n_m-cluster: Aprime n_m-cluster that has been rotated, reflected andtranslated into canonical form
  (Note to web page editor: define the canonicalform here)

• equivalent n_m-cluster: twon_m-clusters are equivalent if and only if they can bemade identical under the combined operations of rotation, reflectionand translation
• nonequivalent n_m-cluster: twon_m-clusters that are not equivalent
  (all primitive n_m-clustersare both prime and nonequivalent)

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

n₂-clusters

An n₂-cluster

is:

• 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

Clusters 1 7 2 Iso

Here is an example of a 6₂-cluster:

The following are the smallest n₂-clusters:

A pictureof thesmallest 6₂-cluster isavailable.

UPDATE:7₂-clustersDO exist!

Chuck Simmons and Landon Curt Noll discovered a number of7₂-clusters on 2006-May-18 15:26:55 PDT:

NOTE: This 7₂-cluster is not be the smallest.The above mentioned 7₂-clusters simply prove thatthese strutures exist.Thesmallest7₂-cluster is:

As recent as 2010-August-26, using an improvedsearch algorithm, 25 nonequivalent (new) 7₂-clusterswere discovered:

Chuck Simmons made images of the 25 7₂-clusters found on2010-August-26 in a blog posting:

Pictures and coordinates of twenty-five recently found 7-clusters

For more information see:

N-Cluster Search: Progress Update (as of 2010-Oct)

It should be noted that Tobias Kreisel and Sascha Kurz reportedthe discovery of two 7-element integral heptagons.While these 7-element integral heptagons are 7 points in R²,no 3 co-linear and no 4 are co-circular and all mutual distancesbetween points are integers > 0, their coordinatesare not lattice points in R².Because some of their points are not lattice points in R²(i.e., some points do not have integer (x,y) coordinates) they arenot 7₂-clusters:

There are Integral Heptagons, No Three Points on a Line, No Four on a Circle

Even so, this above work and results of Kruz and Kreisel is impressiveand is worth reading.

n₃-clusters

An n₃-cluster is:

Clusters 1 7 2 Mods

• 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 circleand 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.

Clusters 1 7 20

Here are some 7₃-clustersrecently discovered by Randall Rathbun:

The following are the smallest n₃-clusters:

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 circleand 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 sphereand 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 andtheorized on the side :-)) found these 7₄-clusterson 8 June 2001:

and this 8₄-clusters on 10 June 2001:

The following are the smallest n₄-clusters:

n₅-clusters

A n₅-cluster is:

Clusters 1 7 2 0

• 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 circleand 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 sphereand 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-sphereand 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-discoveredthese 8₅-clusters:

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

The following are the smallest n₅-clusters:

open n_m-cluster questions, conjectures & observations

n_m-cluster conjectures:

• Erdös/Noll infinite-or-bust n_m-clusterconjecture:
  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-clusterwill be a significant computational challenge.

Open questions about n_m-clusters:

is:

• 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

Clusters 1 7 2 Iso

Here is an example of a 6₂-cluster:

The following are the smallest n₂-clusters:

A pictureof thesmallest 6₂-cluster isavailable.

UPDATE:7₂-clustersDO exist!

Chuck Simmons and Landon Curt Noll discovered a number of7₂-clusters on 2006-May-18 15:26:55 PDT:

NOTE: This 7₂-cluster is not be the smallest.The above mentioned 7₂-clusters simply prove thatthese strutures exist.Thesmallest7₂-cluster is:

As recent as 2010-August-26, using an improvedsearch algorithm, 25 nonequivalent (new) 7₂-clusterswere discovered:

Chuck Simmons made images of the 25 7₂-clusters found on2010-August-26 in a blog posting:

Pictures and coordinates of twenty-five recently found 7-clusters

For more information see:

N-Cluster Search: Progress Update (as of 2010-Oct)

It should be noted that Tobias Kreisel and Sascha Kurz reportedthe discovery of two 7-element integral heptagons.While these 7-element integral heptagons are 7 points in R²,no 3 co-linear and no 4 are co-circular and all mutual distancesbetween points are integers > 0, their coordinatesare not lattice points in R².Because some of their points are not lattice points in R²(i.e., some points do not have integer (x,y) coordinates) they arenot 7₂-clusters:

There are Integral Heptagons, No Three Points on a Line, No Four on a Circle

Even so, this above work and results of Kruz and Kreisel is impressiveand is worth reading.

n₃-clusters

An n₃-cluster is:

Clusters 1 7 2 Mods

• 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 circleand 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.

Clusters 1 7 20

Here are some 7₃-clustersrecently discovered by Randall Rathbun:

The following are the smallest n₃-clusters:

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 circleand 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 sphereand 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 andtheorized on the side :-)) found these 7₄-clusterson 8 June 2001:

and this 8₄-clusters on 10 June 2001:

The following are the smallest n₄-clusters:

n₅-clusters

A n₅-cluster is:

Clusters 1 7 2 0

• 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 circleand 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 sphereand 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-sphereand 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-discoveredthese 8₅-clusters:

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

The following are the smallest n₅-clusters:

open n_m-cluster questions, conjectures & observations

n_m-cluster conjectures:

• Erdös/Noll infinite-or-bust n_m-clusterconjecture:
  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-clusterwill be a significant computational challenge.

Open questions about n_m-clusters:

• No 8₂-cluster has ever been found.Do 8₂-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:

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