Index Page for Daniel W. VanArsdale

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Version: 2/13/024 



Chain Letter Evolution
A history of chain letters based on over 1000 dated examples. Numerically successful variations are identified and analyzed for their replicative advantage. English language paper luck chain letters are the principal topic. Other topics include the Letters from Heaven and the "Media Chain Letter" which lured celebrity participants in 1990. A bibliography on chain letters and pyramid schemes contains hundreds of sources, including all citations in Chain Letter Evolution. A glossary of specialized terminology used in Chain Letter Evolution is also provided.  (Hits 4/2019: 276K)

The Origin of Money Chain Letters
A detailed discussion of the origin and early history of the "Send-a-Dime" craze of 1935 - the first manifestation of money chain letters and pyramid schemes. It is shown that the Send-a-Dime letter was in part modeled on a prior type of luck chain letter. The likely author of Send-a-Dime was an anonymous woman motivated by charity. This constitutes a section of Chain Letter Evolution, but can be read independently.

The Paper Chain Letter Archive
Individual html web pages for over 1000 chain letters arranged by motivational category and date. Categories include luck chain letters (600 examples), money, advocacy, "Letters from Heaven", charity, and exchange. Some unusual items: an anonymous 1917 chain letter giving advice on obtaining conscientious objector status;  a 1920 Sinn Fein revolutionary communication;  
rare scatological parody letters from 1935;  a chain letter invitation to a suicide from 1937;  apparently the only complete text on the web of the famous Circle of Gold pyramid scheme, this from England, 1982;  a libelous Proctor and Gamble boycott alleging satanism from 1986. An annotated index provides easy access to all chain letters in the archive. Chain letter citations in Chain Letter Evolution link to this archive.

Eat No Dynamite *** ADULT CONTENTS ***
A documented collection of over 900 graffiti, including 64 drawings, from the main libraries at UCSB and UCLA. Most were recorded in 1979 and 1980. Comments, including estimates of dispersion.  (Hits 4/2019: 21K)

     See that distant swell?  That is the next great wave of youth rebellion - coming to sweep away the castles of fear.
      Erase racism.
     End the dominance of dominance.
     Some day to walk a distant shore.



Four Letters from Millie: 1930-31
Four personal letters to a male admirer from a woman abandoned by her husband. "Yes dear, I will come there if you send me money."

Matchmaking letter and response: 1917
A matchmaking letter from an engaged woman to a Yale student and his enthusiastic response, "provided if she is musical."

Four Letters to Helen Porter: 1926-31
Four personal letter sent to Miss Helen Porter in Detroit. The first is an unsuccessful proposal of marriage. "I hope you don't think me foolish."


Homogeneous Transformation Matrices
Explicit n-dimensional expressions for homogeneous matrix representations of familiar geometric transformations including projection, dilation, reflection, shear, strain, and rotation. These are extracted from my article "Homogeneous Transformation Matrices for Computer Graphics," Computers & Graphics, Vol. 18, No. 2, 1994. (Hits 4/2019: 148K)

Homogeneous Coordinates: Methods
A selection of facts, theorems and methods utilizing homogeneous coordinates. This is a companion article to "Homogeneous Transformation Matrices" and provides methods sufficient to derive all the matrices presented there. Other topics include: n-dimensional oriented content, intersections, and projection.


The trines - a ternary axiom system 

xxx Note: A detailed exposition, with all proofs, is being prepared. No major errors have been detected, but no comments on the "trines" have been received. (10/29/2023)

The "trines" should prove useful because they provide a simple ternary context to find new theorems and proofs involving familiar fields. 
They may also be useful in characterizing new applications. No advanced methods are used here and results are restricted to a rational venue.

Start with three distinct "trines", A,B,C, and a ternary operation, "junction", the junction of trines E,F,G, denoted by the trine <E,F,G>. Generate all possible expressions using only A,B,C and junction. Here are four examples of trines: B, <B,A,C>, <A,A,C>, and <B,C,<A,<C,A,B>,C>>. The trines are subject to just three equivalence axioms. 

. . . . . . . . . . . . . . . . . . . . . . . . . AXIOMS FOR THE TRINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Generation. The trines are: (1) the distinct primals A,B,C, and (2) the junction of any three trines E,F,G, designated <E,F,G>.
    In each of the following three axioms, a fixed permutation may be applied to the order of the members of every junction in the axiom.  E,F,X,Y,Z are any trines except for the stated restriction.

2. Unpairing.   <E,F,F> = F
3. Inversion.    <E,F,<F,E,X>> = X ,  E and F distinct.
4.  Projection.  <E,F,<X,Y,Z>> = <<E,F,X>,<E,F,Y>,<E,F,Z>>.  

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The following ternary operation on the trines is very useful.  For any trines R,U,V,W the primal replacement in host R by members U,V,W, designated R(U,V,W), is the trine constructed by replacing every A in R by U, every B in R by V, and every C in R by W, simultaneously. Replacements of the form A(U,V,V), B(V,U,V), C(V,V,U), U != V, are undefined. It can be shown by induction that when R is any trine not equal to A, B or C, then as in the axioms, R(E,F,F) = F, R(E,F,R(F,E,X)) = X, and R(E,F,R(X,Y,Z)) = R(R(E,F,X),R(E,F,Y),R(E,F,Z)).

Trines can be constructed within the field Q+(d) of all rational expressions in an indeterminate d, augmented with plus or minus infinity. If the cross ratio [e,f;g,h] = r holds in Q+(d) I write r[e,f,g] = h. Solving the cross ratio [(e-g)(f-h)] / [(e-h)(f-g)] = r for h gives h = r[e,f,g] =  [re(f-g)) + f(g-e)] / [r(f-g) + (g-e)]. Define  r[e,e,e] = e, any r, e. If for the ternary function r[e,f,g] (crossjunction) the members are fixed, a projective function on r is defined. The following notation is useful here: a=plus or minus infinity, b=0, c=1. It is shown that function d[x,y,z] generates every element of Q+(d), starting with inputs a, b, c. Then the correspondences: a~A, 0~B, 1~C, d~D, x+y ~ <<X,B,A>, A, <A,B,Y>>, -x ~  <B, A, <A,<X,B,A>,B>>, xy ~ X(A,B,Y), and 1/x ~ X(B,A,C), provide a bijective mapping of Q+(d) to equivalence classes of trines. The trine axioms correspond to theorems in Q+(d), proving that the trines are relatively consistent with Q+(d).  Defining primal replacement in Q+(d) the same as for trines, the theorem r(e,f,g) = r[e,f,g] is key and may be new. 

Further, the augmented field Q+(d) can be constructed within the trines. The correspondences A~a, B~0, C~1, D=<A,B,C> ~ d and R(E,F,G) ~ r[e,f,g], where R~r, E~e, F~f, G~g, assuming X~x and Y~y, define Q+(d) field operations in terms of trines. These are proved to obey the corresponding field axioms for each operation. This construction shows that the trines have been adequately defined and that they can prove any theorem that Q+(d) can. The trines greatly facilitate proofs by induction.

If for a nonconstant mapping P on any distinct trines E,F,G:  P(<E,F,G>) = <P(E),P(F),P(G)>  P is "projective", and P is called a "projection". Each such mapping  corresponds to a projectivity on Q+(d) (synonymously a "homography" or "linear fractional transformation").  The trines can construct other fields by adding junctional equivalences to the trine axioms. For example, requiring <A,B,C> = <B,A,C> implies d = -1 and generates the rational numbers. Requiring <A,B,C> = <<B,C,A>,A,B> implies d = i = sqrt(-1) and generates the complex numbers with rational coefficients.

The trine axioms are very simple and can construct the positive integers. The product formula x*y = <<a,b,<x,y,c>>, <x,y,<a,b,c>>, c> may then be useful. However trines do not easily manifest the point-line duality of projective geometry, nor do they easily extend to higher geometric dimensions.  





Subject Guide to the Universal Declaration of Human Rights (with drawing "Human Rights Ascendence")

... it is essential, if man is not to be compelled to have recourse, as a last resort, to rebellion against tyranny and oppression, that human rights should be protected by the rule of law,
(Preamble, paragraph 3)

The Self-Appointed Nonviolent Truth Avengers - (SANTA)

Three letters from the "Self Appointed Nonviolent Truth Avengers (SANTA)". These protest: (1) the product placement of Coca Cola in the movie ET (1982), (2) a beef industry TV ad (1987), and (3) the CBS special "Ancient Secrets of the Bible" (1992). SANTA describes itself as follows (1992):
SANTA is a decentralized communication process opposing deceit and exploitation of children by the media. We use only education, rebuke, and satire. No person's name may be mentioned. Anyone who loves the truth and obeys these rules may fully participate in SANTA.

Daniel W. VanArsdale on Twitter
Mathematician, independent researcher. Human Rights, kindness to animals, environmentalism, truth for its own sake, peace, nonviolence, antifundamentalism.

Some of my tweets
Killing people is not the solution to our problems. Killing people is the problem. ( cf. Reagan: "Government is not the solution ...")
The trouble with conservatives is they say they are for freedom but they do not like people to be different.
If there is a devil he is putting war on the table.
How great can America be without honor?
Love is not all we need, but it is what we need most.
When money rules who is free?
At least the secular humanists never burned a witch.
Truth is the goal. The goal is the way. The way is truth.
Creativity favors the Left in music, drama, poetry, and art. But it favors the Right in slander.
Noah had crabs.

Are we great yet? / The universe is made of protons, neutron, electrons, and morons.   Northern Sun t-shirt slogans.
They would rather go down with the Titanic as long as they had first class seats.   Bernie Sanders
I believe this theory that Darwin came up with was ... encouraged by the adversary.   Dr. Ben Carson
American Libertarianism is basically the advocacy of private tyranny.   Noam Chomsky, 2015
Heave an egg out of a Pullman window, and you will hit a Fundamentalist almost anywhere in the Unites States.   H. L. Mencken, 1925.
Those who lack imagination can not imagine what is lacking.   Graffito, Paris, May 1968.
In the future world, only the children will be privileged.   Graffito, Santiago, Chile, 1970.

Every day is Auschwitz for factory animals.  Graffito (edited), from Woman in the John, Susan Shaw, 1979
Republicans have not cared about you since you were a fetus.   Bumper Sticker


The womb gate has opened and closed -- William Blake Dylan
Several poems and fragments given to me by an anonymous hitchhiker in 1973. They express his painful discovery that his mother could no longer help him, and that his counter-culture friendships and drug experiences were fleeting and duality ridden.

Wayne Morris Bell, 1937-1968
A tribute to a close friend who served with me in the Army in Korea in 1963 and was killed in action in Vietnam in 1968.

The Little Golden Concert Hour
Have you ever heard of this local radio program from around 1950, probably broadcast from San Jose, California?

Highly Improbable Personally Experienced Repetitions
A list of 30 experiences, by myself or people close to me, that appear to involve some paranormal phenomenon. Most are seeming precognitive dreams I have had, some recorded in a journal before an improbable repetition occurred. Others were repetitions in daily life. Example:

8/17/1975. Huntington Beach, California (recorded in Oxnard). From journal, volume #3, 6 page stapled insert.
    "Met a woman, Judy, in Huntington Beach whom I was matched with in a computer dating service.  She had a 3 year old boy. Shortly before the child was to go to the baby sitter he began to cry. I went in to his room and got him to talking about his books. He started telling me about his favorite - Donald Duck as a detective. On one page was a picture of 5 or 6 sea animals. He pointed to the picture and named the creatures. But he called a lobster a crab. His mother (on my correcting him) said she was the source of the misidentification.
    "Later I was watching television with his mother. Toward the end of the Cher Show people were dressed as food. A man had a crab around his neck. Cher said something very close to: "it should have been a lobster."

Chess with Chinese Pieces
Advocates free placement of the chess pieces to form a starting position, and the addition of "Chinese" pieces: the Pao or Cannon, Vao and Leo. These move like an orthodox Rook, Bishop and Queen respectively, but must pass over a piece of either color to capture. Presents eleven new chess problems composed by the author using Chinese pieces. For each problem there is a link to its solution. Since uploading this web page in 2004 some computer programs play better than any human. Below are four of the problems presented, with links to their solution.


                            SOLUTION                                                                                      SOLUTION                                                                         SOLUTION    

-   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -                                                                             


Mate in three.  Cannons (paos) on a4, g1, h1.
Daniel W. VanArsdale,  8/2017.


Employ the Dead  

A discussion of Sam Loyd's Dummy Promotion problem.

Daniel W. VanArsdale     email:

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