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Forum for Readers of the Diamond Theory Website
This forum is no longer active. It was intended for readers of the Diamond Theory website, which deals with finite geometry, combinatorics, and group theory. To view the parent website, Diamond Theory, click the following:
Diamond Theory website

Subject: Fraud
Author: Steven H. Cullinane [ Edit \| View ]	Date Posted: 22:39:19 11/12/02 Tue Readers of this site should disregard the derogatory remarks below signed "RT Curtis." This is not the real R. T. Curtis, but an impostor. Deleting the remarks would only tend to confirm the views of their (as yet unknown) author. [ Post a Reply to This Message ]

Subject: Steven H. Cullinane is a Fraud
Author: RT Curtis [ Edit \| View ]	Date Posted: 07:45:54 11/12/02 Tue Steven H. Cullinane has been forging the messages below to save his arse. He cannot explain how he gets the number 322560 from his calculations! RT Curtis [ Post a Reply to This Message ]

[> Subject: Re: Steven H. Cullinane is a Fraud
Author: Steven H. Cullinane [ Edit \| View ]	Date Posted: 06:03:31 11/13/02 Wed As noted elsewhere, this "RT Curtis" is an impostor. [ Post a Reply to This Message ]

Subject: Thank you
Author: Jed Pack [ Edit \| View ]	Date Posted: 22:24:20 11/11/02 Mon Steve Cullinane, Thank you for your help! You are absolutely correct. I should have read your site more carefully. The matter of the number of "configurations" was simply a misunderstanding. I have verified the number you have given. Although I am not a mathematician, I find your work interesting. Jed Pack [ Post a Reply to This Message ]

[> Subject: Re: Thank you
Author: Steven H. Cullinane [ Edit \| View ]	Date Posted: 00:11:25 11/12/02 Tue You're welcome. I'm sorry there was a misunderstanding about the semantics of "configurations." Thanks for your interest, and best wishes. [ Post a Reply to This Message ]

Subject: About Steven Cullinane's "Diamond Theory"

Author:
RT Curtis
[ Edit | View ]

Date Posted: 12:29:15 11/03/02 Sun

The "Diamond Theory" website of Steven Cullinane shows a man who is incapable of telling the truth: a pathological liar who hates and despises the mathematical community; a sociopath caught between the conflicting desires to earn the admiration of mathematicians, and his desire to insult those who ignore him and refuse him his self-perceived due measure of honor and reverie. As such, Steven Cullinane is constantly trying to purchase recognition when he has the funds to advertise on google.com, or steal that recognition by lying and deceiving dmoz.org when money isn't enough. As you can see from the correspondence below, Jed Pack has clearly pointed out serious errors in Steven Cullinane's calculations. Now, instead of admitting that he has been caught with his pants down, Steven Cullinane is questioning Jed Pack's education! Surely, Jed Pack is a more competent mathematician than Steven Cullinane.

[ Post a Reply to This Message ]

[> Subject: Re: About Steven Cullinane's "Diamond Theory"
Author: Steven H. Cullinane [ Edit \| View ]	Date Posted: 08:30:30 11/10/02 Sun Isn't free speech wonderful? I am curious about two things... (1) Are you the R. T. Curtis of Miracle Octad Generator fame -- and if so, why is your email address not his academic email address, and (2) On what, other than the evidence shown in this forum, do you base your assessment of Pack's abilities? [ Post a Reply to This Message ]

[> [> Subject: Re: About Steven Cullinane's "Diamond Theory"

Author:
Robert T. Curtis
[ Edit | View ]

Date Posted: 12:11:12 11/11/02 Mon

I wish to make it clear that I had nothing whatsoever to
do with the message posted on this web-site as having
been sent by R.T.Curtis. There may, of course, be another
R.T.Curtis around who has a particular grudge against
Professor Steven Cullinane. I do not. Indeed, the exchanges
we have had with one another over the years have always
been cordial and civilised.

If someone is deliberately using my name to attack Steven
Cullinane anonymously, it shows malice and cowardice
unusual in the mathematical world.

Robert Curtis.

[ Post a Reply to This Message ]

Subject: Four by Four case

Author:
Jed Pack
[ Edit | View ]

Date Posted: 20:01:10 07/10/02 Wed

Mr. Cullinane,

There are exactly 840 unique configurations that can be attained through swapping columns, rows, or quadrants in the 4 by 4 case. As justification of the above statement, consider the following:

I programmed a computer to perform repeated random column, row, and quadrant swaps and keep track of unique configurations generated in this manner. After several thousand iterations, there were 840 unique configurations. This only proves that there are AT LEAST 840 members of the group. To prove there are EXACTLY 840 members of the group, I performed each of the 18 transformations (swapping the 6 pairs of rows, columns and quadrants) on each of the 840 configurations. In each case the resulting configuration was one of the 840 configurations. This proves that the 840 configurations are a group and that they are the only configurations that can be attained through the above mentioned transformations.

P.S. In my earlier message I stated that there were AT MOST 900. The previous argument is valid. The following addendum shows more formally that there are at most 840:

The two binary matricies that describe the configuration (call them A and B) are always different and never complementary (i.e. A is not an element of the set {B, 1-B}). This statement follows inductively from two facts:
1) The condition is obviously true at the beginning.
2) No transformation can change this since the same operation is performed on both A and B.

As a result, 60 of the 900 configurations described by arbitrary combinations of two of the thirty available matricies are impossible. This leaves 840 possible configurations.

P.P.S. If you are interested, I can send you data files that describe each configuration. I can also include a 840 by 18 matrix indicating which configuration is produced by performing each of the 18 transformations on each of the 840 configurations.

P.P.P.S. The two binary matricies mentioned in my previous posting DO fully describe the configuration. Each of the 16 cells has four possible states (black corner can be top right, top left, bottom right, or bottom left). Thus, the original configuration (with four black diamonds) can clearly be described by one matrix as follows:
-------------
| 2 | 1 | 2 | 1 |
| 4 | 3 | 4 | 3 |
| 2 | 1 | 2 | 1 |
| 4 | 3 | 4 | 3 |
-------------
where a 1 means the black corner is bottom left, 2 means bottom right, 3 means top left, and 4 means top right.

This matrix can easily be generated using the two binary matricies as follows: 2*A+B+1.

As a result, these two matricies also fully describe the configuration.

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[> Subject: Re: Four by Four case

Author:
S. H. Cullinane
[ Edit | View ]

Date Posted: 19:51:04 07/18/02 Thu

Dear Mr. Pack:

Congratulations on your realization that there are EXACTLY 840 of what you call "configurations." This is exactly right, and it is what I meant by the following sentence in my Diamond Theory site --

"The 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2)."

A careful reader of my site would have found this in my remarks on the 4x4 case of the diamond theorem. I have no use for a file illustrating the 840 configurations, since they are easily obtained by substituting the 4 design elements in any of 4! = 24 ways in each of the 35 structures of the configurations that are illustrated in my Research Note 3, "Orthogonality of Latin Squares Viewed as Skewness of Lines." This is why I wrote "840 = 35 x 24."

It is clear from your statement that "the 840 configurations are a group" that you do not know what a group, in the technical mathematical sense, is.

I suggest you find out. Don't they teach mathematics at Brigham Young University?

By the way, "matricies" is usually spelled "matrices."

[ Post a Reply to This Message ]

Subject: Number of four by four permutations...

Author:
Jed Pack
[ Edit | View ]

Date Posted: 18:35:25 04/17/02 Wed

Steven Cullinane,

Your website states:

In the 4x4 case, D is a four-diamond figure (left, below) and G is a group of 322,560 permutations
generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the
four 2x2 quadrants. Every G-image of D (as at right, below) has some ordinary or color-interchange
symmetry.

I wonder how you got the number 322,560.

I suggest the number is at most 30^2=900.
Please let me know if you find the error in my logic.

In the original four diamond configuration. The matrix indicating which squares have the black corner
either at top-right or top-left is:

0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1

Similarly, the matrix indicating which squares have the black corner either at top-right or bottom-right
is:

1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0

These two matricies fully describe the configuration. Swapping columns, rows, and quadrants can change
these matricies into any of the following other 30 matricies (there are no other possibilities):

0 0 1 1 | 0 1 1 0 | 0 0 1 1 | 1 1 1 1 | 1 0 1 0 |
0 0 1 1 | 0 1 1 0 | 1 1 0 0 | 0 0 0 0 | 1 0 1 0 |
0 0 1 1 | 1 0 0 1 | 1 1 0 0 | 0 0 0 0 | 0 1 0 1 |
0 0 1 1 | 1 0 0 1 | 0 0 1 1 | 1 1 1 1 | 0 1 0 1 |

0 1 0 1 | 0 0 0 0 | 0 1 0 1 | 1 0 0 1 | 1 1 0 0 |
0 1 0 1 | 1 1 1 1 | 1 0 1 0 | 0 1 1 0 | 1 1 0 0 |
0 1 0 1 | 0 0 0 0 | 1 0 1 0 | 1 0 0 1 | 0 0 1 1 |
0 1 0 1 | 1 1 1 1 | 0 1 0 1 | 0 1 1 0 | 0 0 1 1 |

0 1 1 0 | 0 0 1 1 | 0 1 1 0 | 1 0 1 0 | 1 1 1 1 |
0 1 1 0 | 1 1 0 0 | 1 0 0 1 | 0 1 0 1 | 1 1 1 1 |
0 1 1 0 | 0 0 1 1 | 1 0 0 1 | 1 0 1 0 | 0 0 0 0 |
0 1 1 0 | 1 1 0 0 | 0 1 1 0 | 0 1 0 1 | 0 0 0 0 |

0 0 0 0 | 0 1 0 1 | 1 0 0 1 | 1 1 0 0 | 1 0 0 1 |
0 0 0 0 | 1 0 1 0 | 0 1 1 0 | 0 0 1 1 | 1 0 0 1 |
1 1 1 1 | 0 1 0 1 | 0 1 1 0 | 1 1 0 0 | 1 0 0 1 |
1 1 1 1 | 1 0 1 0 | 1 0 0 1 | 0 0 1 1 | 1 0 0 1 |

0 0 1 1 | 0 1 1 0 | 1 0 1 0 | 1 1 1 1 | 1 0 1 0 |
0 0 1 1 | 1 0 0 1 | 0 1 0 1 | 0 0 0 0 | 1 0 1 0 |
1 1 0 0 | 0 1 1 0 | 0 1 0 1 | 1 1 1 1 | 1 0 1 0 |
1 1 0 0 | 1 0 0 1 | 1 0 1 0 | 0 0 0 0 | 1 0 1 0 |

0 1 0 1 | 0 0 0 0 | 1 1 0 0 | 1 0 0 1 | 1 1 0 0 |
0 1 0 1 | 1 1 1 1 | 0 0 1 1 | 1 0 0 1 | 1 1 0 0 |
1 0 1 0 | 1 1 1 1 | 0 0 1 1 | 0 1 1 0 | 1 1 0 0 |
1 0 1 0 | 0 0 0 0 | 1 1 0 0 | 0 1 1 0 | 1 1 0 0 |

Consequently, any pattern that can be obtained through such transformations can be described by a pair of
these two matricies.

There are 30^2 such pairs, and hence can be no more than 30^2 patterns obtained through the mentioned
transformations.

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[> Subject: Re: Number of four by four permutations...

Author:
S. H. Cullinane
[ Edit | View ]

Date Posted: 00:16:40 04/25/02 Thu

Dear Mr. Pack:

Here's how I got the number 322,560:
"G is isomorphic to the affine group A on the linear 4-space over GF(2)," as stated in my website.
I suggest you consult a book on finite geometry and group theory such as Geometry and Symmetry, by Paul Yale, to learn more about what the above sentence means.
It is a nontrivial exercise to PROVE the sentence.
Your error seems to be in your statement that "these two matrices fully describe the configuration." Try describing the configuration with two orthogonal 4x4 Latin squares instead.

Yours truly, S. H. Cullinane

[ Post a Reply to This Message ]

Subject: What is the point of diamond theory
Author: Bob [ Edit \| View ]	Date Posted: 18:12:04 05/10/01 Thu What is the point of diamond theory? The pictures are pretty, but there are lots of pretty things to research in mathematics. Why is diamond theory more important than the other fields? Bob. [ Post a Reply to This Message ]

[> Subject: Re: What is the point of diamond theory

Author:
Steve Cullinane
[ Edit | View ]

Date Posted: 19:59:52 05/11/01 Fri

Dear Bob:

You ask what is the point of diamond theory.

Answer 1:

"...that profoundly serene and satisfying quality
which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms."
- Edward Sapir, "The Grammarian and his Language," American Mercury 1:149-155, 1924

Answer 2:

As Weyl pointed out, symmetry is one of the most important concepts in mathematics (not to mention physics). Symmetry is often defined as invariance of some property under a group of transformations. Diamond theory is of aesthetic and pedagogical interest because what is invariant under its groups of tranformations is.... symmetry itself!

Answer 3:

Diamond theory provides insight (via the MOG of R. T. Curtis) into the structure of M24 -- the most interesting finite group, according to J. H. Conway.

I don't claim that diamond theory is more important than other fields, but that it forms a natural part of the fields of finite geometry, finite group theory, and combinatorics.

Yours truly, Steve Cullinane

[ Post a Reply to This Message ]

Subject: Doyou have any history on "The Starr Of The Mountain" Crown Jewel of Afganastan
Author: Al Burness [ Edit \| View ]	Date Posted: 20:27:56 05/05/01 Sat I understand that this Jewel is one of the diamonds in one of the crowns of England. Where do I find more history about this rare diamond. [ Post a Reply to This Message ]

[> Subject: Re: Doyou have any history on "The Starr Of The Mountain" Crown Jewel of Afganastan
Author: S. H. Cullinane [ Edit \| View ]	Date Posted: 20:34:32 05/08/01 Tue Dear Mr. Burness: Sorry, the diamond theory website is about mathematics, not rocks. Best wishes for your search, Steve Cullinane P.S. - Try Google.com search engine - It seems to be the best. [ Post a Reply to This Message ]