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Showing posts with label Euler. Show all posts
Showing posts with label Euler. Show all posts

Wednesday, May 16, 2012

Euler Diagram



An Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is disjoint (has no members in common) with "animals".
An Euler diagram is a diagrammatic means of representing sets and their relationships. The first use of "Eulerian circles" is commonly attributed to Swiss mathematician Leonhard Euler (1707–1783). They are closely related to Venn diagrams.

Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since then, they have also been adopted by other curriculum fields such as reading.[1]

Contents

Overview

Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. The sizes or shapes of the curves are not important: the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness).
Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it.


Examples of small Venn diagrams (on left) with shaded regions representing empty sets, showing how they can be easily transformed into equivalent Euler diagrams (right).
Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all the possible zones of overlap between its curves, representing all combinations of inclusion/exclusion of its constituent sets, but in an Euler diagram some zones might be missing. When the number of sets grows beyond 3, or even with three sets, but under the allowance of more than two curves passing at the same point, we start seeing the appearance of multiple mathematically unique Venn diagrams. Venn diagrams represent the relationships between n sets, with 2n zones, Euler diagrams may not have all zones. (An example is given below in the History section; in the top-right illustration the O and I diagrams are merely rotated; Venn stated that this difficulty in part led him to develop his diagrams).

In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a universe of discourse. In the examples above, the Euler diagram depicts that the sets Animal and Mineral are disjoint since the corresponding curves are disjoint, and also that the set Four Legs is a subset of the set of Animals. The Venn diagram, which uses the same categories of Animal, Mineral, and Four Legs, does not encapsulate these relationships. Traditionally the emptiness of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent emptiness either by shading or by the use of a missing region.
Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the diagram to the right, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs.
 

History

 


Photo of page from Hamilton's 1860 "Lectures" page 180. (Click on it, up to two times, to enlarge). The symbolism A, E, I, and O refer to the four forms of the syllogism. The small text to the left says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise."

On the right is a photo of page 74 from Couturat 1914 wherein he labels the 8 regions of the Venn diagram. The modern name for these "regions" is minterms. These are shown on the left with the variables x, y and z per Venn's drawing. The symbolism is as follows: logical AND ( & ) is represented by arithmetic multiplication, and the logical NOT ( ~ )is represented by " ' " after the variable, e.g. the region x'y'z is read as "NOT x AND NOT y AND z" i.e. ~x & ~y & z.

Both the Veitch and Karnaugh diagrams show all the minterms, but the Veitch is not particularly useful for reduction of formulas. Observe the strong resemblance between the Venn and Karnaugh diagrams; the colors and the variables x, y, and z are per Venn's example.
As shown in the illustration to the right, Sir William Hamilton in his posthumously published Lectures on Metaphysics and Logic (1858–60) asserts that the original use of circles to "sensualize ... the abstractions of Logic" (p. 180) was not Leonhard Paul Euler (1707–1783) but rather Christian Weise (?–1708) in his Nucleus Logicoe Weisianoe that appeared in 1712 posthumously. He references Euler's Letters to a German Princess on different Matters of Physics and Philosophy1" [1Partie ii., Lettre XXXV., ed. Cournot. – ED.][2]
In Hamilton's illustration the four forms of the syllogism as symbolized by the drawings A, E, I and O are:[3]
  • A: The Universal Affirmative, Example: "All metals are elements".
  • E: The Universal Negative, Example: "No metals are compound substances".
  • I: The Particular Affirmative, Example: "Some metals are brittle".
  • O: The Particular Negative, Example: "Some metals are not brittle".
In his 1881 Symbolic Logic Chapter V "Diagrammatic Representation", John Venn (1834–1923) comments on the remarkable prevalence of the Euler diagram:
"...of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose:-somewhat at random, as they happened to be most accessible :-it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian Scheme." (Footnote 1 page 100)

Composite of two pages 115–116 from Venn 1881 showing his example of how to convert a syllogism of three parts into his type of diagram. Venn calls the circles "Eulerian circles" (cf Sandifer 2003, Venn 1881:114 etc) in the "Eulerian scheme" (Venn 1881:100) of "old-fashioned Eulerian diagrams" (Venn 1881:113).
But nevertheless, he contended "the inapplicability of this scheme for the purposes of a really general Logic" (page 100) and in a footnote observed that "it fits in but badly even with the four propositions of the common Logic [the four forms of the syllogism] to which it is normally applied" (page 101). Venn ends his chapter with the observation that will be made in the examples below – that their use is based on practice and intuition, not on a strict algorithmic practice:
“In fact ... those diagrams not only do not fit in with the ordinary scheme of propositions which they are employed to illustrate, but do not seem to have any recognized scheme of propositions to which they could be consistently affiliated.” (pp. 124–125)
Finally, in his Chapter XX HISTORIC NOTES Venn gets to a crucial criticism (italicized in the quote below); observe in Hamilton's illustration that the O (Particular Negative) and I (Particular Affirmative) are simply rotated:
"We now come to Euler's well-known circles which were first described in his Lettres a une Princesse d'Allemagne (Letters 102–105). The weak point about these consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions.... This defect must have been noticed from the first in the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well". (italics added: page 424)
(Sandifer 2003 reports that Euler makes such observations too; Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations). Whatever the case, armed with these observations and criticisms, Venn then demonstrates (pp. 100–125) how he derived what has become known as his Venn diagrams from the "old-fashioned Euler diagrams". In particular he gives an example, shown on the left.
By 1914 Louis Couturat (1868–1914) had labeled the terms as shown on the drawing on the right. Moreover, he had labeled the exterior region (shown as a'b'c') as well. He succinctly explains how to use the diagram – one must strike out the regions that are to vanish:
"VENN'S method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only strike out (by shading) those which are made to vanish by the data of the problem." (italics added p. 73)
Given the Venn's assignments, then, the unshaded areas inside the circles can be summed to yield the following equation for Venn's example:
"No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z for the unshaded area inside the circles (but note that this is not entirely correct; see the next paragraph).
In Venn the 0th term, x'y'z', i.e. the background surrounding the circles, does not appear. Nowhere is it discussed or labeled, but Couturat corrects this in his drawing. The correct equation must include this unshaded area shown in boldface:
"No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z + x'y'z' .
In modern usage the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the domain of discourse.
Couturat now observes that, in a direct algorithmic (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "No X is Z". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems":
"It does not show how the data are exhibited by canceling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i. e., "throwing of the problem into an equation" and the transformation of the premises, nor with the subsequent steps, i. e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form."(p. 75)
Thus the matter would rest until 1952 when Maurice Karnaugh (1924– ) would adapt and expand a method proposed by Edward W. Veitch; this work would rely on the truth table method precisely defined in Emil Post's 1921 PhD thesis "Introduction to a general theory of elementary propositions" and the application of propositional logic to switching logic by (among others) Claude Shannon, George Stibitz, and Alan Turing.[4] For example, in chapter "Boolean Algebra" Hill and Peterson (1968, 1964) present sections 4.5ff "Set Theory as an Example of Boolean Algebra" and in it they present the Venn diagram with shading and all. They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement:
"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6." (p. 64)
In Chapter 6, section 6.4 "Karnaugh Map Representation of Boolean Functions" they begin with:
"The Karnaugh map1 [1Karnaugh 1953] is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram." (pp. 103–104)
The history of Karnaugh's development of his "chart" or "map" method is obscure. Karnaugh in his 1953 referenced Veitch 1951, Veitch referenced Claude E. Shannon 1938 (essentially Shannon's Master's thesis at M.I.T.), and Shannon in turn referenced, among other authors of logic texts, Couturat 1914. In Veitch's method the variables are arranged in a rectangle or square; as described in Karnaugh map, Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) a hypercube.

Example: Euler- to Venn-diagram and Karnaugh map

This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "No X's are Z's". In the illustration and table the following logical symbols are used:
1 can be read as "true", 0 as "false"
~ for NOT and abbreviated to ' when illustrating the minterms e.g. x' =defined NOT x,
+ for Boolean OR (from Boolean algebra: 0+0=0, 0+1 = 1+0 = 1, 1+1=1)
& (logical AND) between propositions; in the mintems AND is omitted in a manner similar to arithmetic multiplication: e.g. x'y'z =defined ~x & ~y & z (From Boolean algebra: 0*0=0, 0*1 = 1*0=0, 1*1 = 1, where * is shown for clarity)
→ (logical IMPLICATION): read as IF ... THEN ..., or " IMPLIES ", P → Q =defined NOT P OR Q

Before it can be presented in a Venn diagram or Karnaugh Map, the Euler diagram's syllogism "No Y is Z, All X is Y" must first be reworded into the more formal language of the propositional calculus: " 'It is not the case that: Y AND Z' AND 'If an X then a Y' ". Once the propositions are reduced to symbols and a propositional formula ( ~(y & z) & (x → y) ), one can construct the formula's truth table; from this table the Venn and/or the Karnaugh map are readily produced. By use of the adjacency of "1"s in the Karnaugh map (indicated by the grey ovals around terms 0 and 1 and around terms 2 and 6) one can "reduce" the example's Boolean equation i.e. (x'y'z' + x'y'z) + (x'yz' + xyz') to just two terms: x'y' + yz'. But the means for deducing the notion that "No X is Z", and just how the reduction relates to this deduction, is not forthcoming from this example.
Given a proposed conclusion such as "No X is a Z", one can test whether or not it is a correct deduction by use of a truth table. The easiest method is put the starting formula on the left (abbreviate it as "P") and put the (possible) deduction on the right (abbreviate it as "Q") and connect the two with logical implication i.e. P → Q, read as IF P THEN Q. If the evaluation of the truth table produces all 1's under the implication-sign (→, the so-called major connective) then P → Q is a tautology. Given this fact, one can "detach" the formula on the right (abbreviated as "Q") in the manner described below the truth table.
Given the example above, the formula for the Euler and Venn diagrams is:
"No Y's are Z's" and "All X's are Y's": ( ~(y & z) & (x → y) ) =defined P
And the proposed deduction is:
"No X's are Z's": ( ~ (x & z) ) =defined Q
So now the formula to be evaluated can be abbreviated to:
( ~(y & z) & (x → y) ) → ( ~ (x & z) ): P → Q
IF ( "No Y's are Z's" and "All X's are Y's" ) THEN ( "No X's are Z's" )
The Truth Table demonstrates that the formula ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) is a tautology as shown by all 1's in yellow column..
Square # Venn, Karnaugh region
x y z
(~ (y & z) & (x y)) (~ (x & z))
0 x'y'z' 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0
1 x'y'z 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1
2 x'yz' 0 1 0 1 1 0 0 1 0 1 1 1 1 0 0 0
3 x'yz 0 1 1 0 1 1 1 0 0 1 1 1 1 0 0 1
4 xy'z' 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0
5 xy'z 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 1
6 xyz' 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0
7 xyz 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1
At this point the above implication P → Q (i.e. ~(y & z) & (x → y) ) → ~(x & z) ) is still a formula, and the deduction – the "detachment" of Q out of P → Q – has not occurred. But given the demonstration that P → Q is tautology, the stage is now set for the use of the procedure of modus ponens to "detach" Q: "No X's are Z's" and dispense with the terms on the left.[5]
Modus ponens (or "the fundamental rule of inference"[6]) is often written as follows: The two terms on the left, "P → Q" and "P", are called premises (by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called the conclusion:
P → Q, P ⊢ Q
For the modus ponens to succeed, both premises P → Q and P must be true. Because, as demonstrated above the premise P → Q is a tautology, "truth" is always the case no matter how x, y and z are valued, but "truth" will only be the case for P in those circumstances when P evaluates as "true" (e.g. rows 0 OR 1 OR 2 OR 6: x'y'z' + x'y'z + x'yz' + xyz' = x'y' + yz').[7]
P → Q , P ⊢ Q
i.e.: ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) , ( ~(y & z) & (x → y) ) ⊢ ( ~ (x & z) )
i.e.: IF "No Y's are Z's" and "All X's are Y's" THEN "No X's are Z's", "No Y's are Z's" and "All X's are Y's" ⊢ "No X's are Z's"
One is now free to "detach" the conclusion "No X's are Z's", perhaps to use it in a subsequent deduction (or as a topic of conversation).
The use of tautological implication means that other possible deductions exist besides "No X's are Z's"; the criterion for a successful deduction is that the 1's under the sub-major connective on the right include all the 1's under the sub-major connective on the left (the major connective being the implication that results in the tautology). For example, in the truth table, on the right side of the implication (→, the major connective symbol) the bold-face column under the sub-major connective symbol " ~ " has the all the same 1s that appear in the bold-faced column under the left-side sub-major connective & (rows 0, 1, 2 and 6), plus two more (rows 3 and 4).

Gallery

Footnotes

  1. ^ Strategies for Reading Comprehension Venn Diagrams
  2. ^ By the time these lectures of Hamilton were published, Hamilton too had died. His editors (symbolized by ED.), responsible for most of the footnoting, were the logicians Henry Longueville Mansel and John Veitch.
  3. ^ Hamilton 1860:179. The examples are from Jevons 1881:71ff.
  4. ^ See footnote at George Stibitz.
  5. ^ This is a sophisticated concept. Russell and Whitehead (2nd edition 1927) in their Principia Mathematica describe it this way: "The trust in inference is the belief that if the two former assertions [the premises P, P→Q ] are not in error, the final assertion is not in error . . . An inference is the dropping of a true premiss [sic]; it is the dissolution of an implication" (p. 9). Further discussion of this appears in "Primitive Ideas and Propositions" as the first of their "primitive propositions" (axioms): *1.1 Anything implied by a true elementary proposition is true" (p. 94). In a footnote the authors refer the reader back to Russell's 1903 Principles of Mathematics §38.
  6. ^ cf Reichenbach 1947:64
  7. ^ Reichenbach discusses the fact that the implication P → Q need not be a tautology (a so-called "tautological implication"). Even "simple" implication (connective or adjunctive) will work, but only for those rows of the truth table that evaluate as true, cf Reichenbach 1947:64–66.

References

By date of publishing:
  • Sir William Hamilton 1860 Lectures on Metaphysics and Logic edited by Henry Longueville Mansel and John Veitch, William Blackwood and Sons, Edinburgh and London.
  • W. Stanley Jevons 1880 Elemetnary Lessons in Logic: Deductive and Inductive. With Copious Questions and Examples, and a Vocabulary of Logical Terms, M. A. MacMillan and Co., London and New York.
  • John Venn 1881 Symbolic Logic, MacMillan and Co., London.
  • Alfred North Whitehead and Bertrand Russell 1913 1st edition, 1927 2nd edition Principia Mathematica to *56 Cambridge At The University Press (1962 edition), UK, no ISBN.
  • Louis Couturat 1914 The Algebra of Logic: Authorized English Translation by Lydia Gillingham Robinson with a Preface by Philip E. B. Jourdain, The Open Court Publishing Company, Chicago and London.
  • Emil Post 1921 "Introduction to a general theory of elementary propositions" reprinted with commentary by Jean van Heijenoort in Jean van Heijenoort, editor 1967 From Frege to Gödel: A Sourcebook of Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, ISBN 0-674-42449-8 (pbk.)
  • Claude E. Shannon 1938 "A Symbolic Analysis of Relay and Switching Circuits", Transactions American Institute of Electrical Engineers vol 57, pp. 471–495. Derived from Claude Elwood Shannon: Collected Papers edited by N.J.A. Solane and Aaron D. Wyner, IEEE Press, New York.
  • Hans Reichenbach 1947 Elements of Symbolic Logic republished 1980 by Dover Publications, Inc., NY, ISBN 0-486-24004-5.
  • Edward W. Veitch 1952 "A Chart Method for Simplifying Truth Functions", Transactions of the 1952 ACM Annual Meeting, ACM Annual Conference/Annual Meeting "Pittsburgh", ACM, NY, pp. 127–133.
  • Maurice Karnaugh November 1953 The Map Method for Synthesis of Combinational Logic Circuits, AIEE Committee on Technical Operations for presentation at the AIEE summer General Meeting, Atlantic City, N. J., June 15–19, 1953, pp. 593–599.
  • Frederich J. Hill and Gerald R. Peterson 1968, 1974 Introduction to Switching Theory and Logical Design, John Wiley & Sons NY, ISBN 0-71-39882-9.
  • Ed Sandifer 2003 How Euler Did It, http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf

External links

Thursday, November 19, 2009

Coffee and Donut?




A continuous deformation (homeomorphism) of a coffee cup into a doughnut (torus) and back.
Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick."
***
This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vectorfield on the sphere. As with the Bridges of Königsberg, the result does not depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind of smooth blob, as long as it has no holes.

In order to deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.

Intuitively two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist can't distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. A precise definition of homeomorphic, involving a continuous function with a continuous inverse, is necessarily more technical.

Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.
Equivalence classes of the English alphabet in uppercase sans-serif font (Myriad); left - homeomorphism, right - homotopy equivalence



 


An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends partially on the font used. The figures use a sans-serif font named Myriad.

Notice that homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several of the homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent, e.g. O fits inside P and the tail of the P can be squished to the "hole" part.

Thus, the homeomorphism classes are: one hole two tails, two holes no tail, no holes, one hole no tail, no holes three tails, a bar with four tails (the "bar" on the K
is almost too short to see), one hole one tail, and no holes four tails.
The homotopy classes are larger, because the tails can be squished down to a point. The homotopy classes are: one hole, two holes, and no holes.

To be sure we have classified the letters correctly, we not only need to show that two letters in the same class are equivalent, but that two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by suitably selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.

Letter topology has some practical relevance in stencil typography. The font Braggadocio, for instance, has stencils that are made of one connected piece of material.

Friday, January 02, 2009

The Whole World is a Stage

Euler product formula


Now you must know what sets my mind to think in such abstract spaces. "Probability of seeing a stage in a concert."

All The World's A Stageby William Shakespeare
From: As you Like It, Act II Scene VII

Jaques:All the world's a stage,
And all the men and women merely players:
They have their exits and their entrances;
And one man in his time plays many parts,
His acts being seven ages. At first the infant,
Mewling and puking in the nurse's arms.
And then the whining school-boy, with his satchel
And shining morning face, creeping like snail
Unwillingly to school. And then the lover,
Sighing like furnace, with a woeful ballad
Made to his mistress' eyebrow. Then a soldier,
Full of strange oaths and bearded like the pard,
Jealous in honour, sudden and quick in quarrel,
Seeking the bubble reputation
Even in the cannon's mouth. And then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part. The sixth age shifts
Into the lean and slipper'd pantaloon,
With spectacles on nose and pouch on side,
His youthful hose, well saved, a world too wide
For his shrunk shank; and his big manly voice,
Turning again toward childish treble, pipes
And whistles in his sound. Last scene of all,
That ends this strange eventful history,
Is second childishness and mere oblivion,
Sans teeth, sans eyes, sans taste, sans everything.


So of course I am in this space of a kind looking and trying to orientate to watch the performance. My position to the stage, from the stage to myself. Whose to think such formulas would provide a solid description of the effort? So now I am embroiled in information of all kinds here. Shakespeare plays on.

"Liesez Euler, Liesez Euler, c'est notre maître à tous"
("Read Euler, read Euler, he is our master in everything")
- Laplace


The world can be a interesting place once you see it's multi-dimensional ability to have more information then what is apparent around us. We have to open our eyes and listen more carefully. Are you listening Glaucon?:)

"Dyson, one of the most highly-regarded scientists of his time, poignantly informed the young man that his findings into the distribution of prime numbers corresponded with the spacing and distribution of energy levels of a higher-ordered quantum state." Mathematics Problem That Remains Elusive—And Beautiful By Raymond Petersen


So in general such a space when held to the "thinking of points" what is it that shall gauge the thinking mind to think it is possible to explain itself "as gaps within the apparent world" of the everyday? Shall every person care when they are embroiled within the business of the media reported? Do you think the person next to you does not care about the world? Do you not think they experience? The voice is cast from the stage and all is heard in it's reverberations. No head involved, just the bouncing and measure of distance, in an echo of reason. Does an elemental thought have no substance?

Prime Numbers Get Hitched by Marcus du Sautoy • Posted March 27, 2006 12:40 AM

It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons. But mathematicians are very suspicious of statistics. We like things to be exact. Keating and Snaith had used physics to make a very precise prediction that left no room for the power of statistics to see patterns where there are none.

Mathematicians are now convinced. That chance meeting in the common room in Princeton resulted in one of the most exciting recent advances in the theory of prime numbers. Many of the great problems in mathematics, like Fermat's Last Theorem, have only been cracked once connections were made to other parts of the mathematical world. For 150 years many have been too frightened to tackle the Riemann Hypothesis. The prospect that we might finally have the tools to understand the primes has persuaded many more mathematicians and physicists to take up the challenge. The feeling is in the air that we might be one step closer to a solution. Dyson might be right that the opportunity was missed to discover relativity 40 years earlier, but who knows how long we might still have had to wait for the discovery of connections between primes and quantum physics had mathematicians not enjoyed a good chat over tea.




It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.). See more info on Mersenne Prime.




So of course, how many ways can one travel to get too?

The river Pregel divides the town of Konigsberg into four separate land masses, A, B, C, and D. Seven bridges connect the various parts of town, and some of the town's curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. All who tried ended up in failure, including the Swiss mathematician, Leonhard Euler (1707-1783)(pronounced "oiler"), a notable genius of the eighteenth-century.

As a lay person being introduced to the strange world of mathematics it is always interesting to me in the way one can see in abstract processes.

The Bridges of KonigsbergThe Beginnings of Topology...The Generalization to Graph Theory
Euler generalized this mode of thinking by making the following definitions and proving a theorem:

Definition: A network is a figure made up of points (vertices) connected by non-intersecting curves (arcs).

Definition: A vertex is called odd if it has an odd number of arcs leading to it, other wise it is called even.

Definition: An Euler path is a continuous path that passes through every arc once and only once.

Theorem: If a network has more than two odd vertices, it does not have an Euler path.

Euler also proved the converse:

Theorem: If a network has two or less odd vertices, it has at least one Euler path.

Friday, October 05, 2007

Euler's Konigsberg's Bridges Problem

"Liesez Euler, Liesez Euler, c'est notre maître à tous"
("Read Euler, read Euler, he is our master in everything") -
Laplace


I should say here that the post by Guest post: Marni D. Sheppeard, “Is Category Theory Useful ?” over at A Quantum Diaries Survivor, continues to invoke my minds journey into the abstract spaces of mathematics.

The river Pregel divides the town of Konigsberg into four separate land masses, A, B, C, and D. Seven bridges connect the various parts of town, and some of the town's curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. All who tried ended up in failure, including the Swiss mathematician, Leonhard Euler (1707-1783)(pronounced "oiler"), a notable genius of the eighteenth-century.

As a lay person being introduced to the strange world of mathematics it is always interesting to me in the way one can see in abstract processes.

The Beginnings of Topology...The Generalization to Graph Theory
Euler generalized this mode of thinking by making the following definitions and proving a theorem:

Definition: A network is a figure made up of points (vertices) connected by non-intersecting curves (arcs).

Definition: A vertex is called odd if it has an odd number of arcs leading to it, other wise it is called even.

Definition: An Euler path is a continuous path that passes through every arc once and only once.

Theorem: If a network has more than two odd vertices, it does not have an Euler path.

Euler also proved the converse:

Theorem: If a network has two or less odd vertices, it has at least one Euler path.


Leonhard Paul Euler (pronounced Oiler; IPA [ˈɔʏlɐ]) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. He published more papers than any other mathematician of his time.[2]

Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[3] He is also renowned for his work in mechanics, optics, and astronomy.


Portrait by Johann Georg Brucker- Born April 15, 1707(1707-04-15)Basel, Switzerland and Died September 18 [O.S. September 7] 1783
St Petersburg, Russia


You have to understand that as a lay person, my education is obtained through the internet. This is not without years of study(many books) in a lot of areas, that I could be said I am in a profession of anything, other then the student, who likes to learn a lot.

To find connections between the "real world" and what a lot think of as "to abstract to be real."

Any such expansionary mode of thinking, if not understood, as in the Case of Riemann's hypothesis seen in relation to Ulam's Spiral, one might have never understood the use of "Pascal's triangle" as well.

These are "base systems of mathematics" that are describing processes in nature?

See:Euler - 300th anniversay lecture

Friday, March 23, 2007

Lingua Cosmica

It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.). See more info on Mersenne Prime.

I always find it interesting that the ability of the mind to do it's gymnastics, had to have some "background information" with which we could assign "the acrobatics of thinking" to special sequences. Thus create some commonality of exchange.

Might we think the computerized world will give us an "human emotive side of being."

See here for Against Symmetry explanation.

So born from it's "original position" what asymmetry was produced to have the universe have it's special way with which it will deal with it's inhabitants? Any "point source" has a greater potential and from a "perfect symmetry" you had to know where this existed?

Lee Smolin will then lead you away from perfect symmetry and explain why?

G -> H -> ... -> SU(3) x SU(2) x U(1) -> SU(3) x U(1)

Here, each arrow represents a symmetry breaking phase transition where matter changes form and the groups - G, H, SU(3), etc. - represent the different types of matter, specifically the symmetries that the matter exhibits and they are associated with the different fundamental forces of nature


So why not think for a minute that if you had "crossed wires" how might you see the world and think, how strange a Synesthesist to have such "emotive reactions instantaneously" bring forth perceived coloured responses. Colours perhaps, as diverse as the Colour of Gravity?

How much of a joke shall I play with peoples minds to think the choice of the observer has consequences? That those consequences are indeed coloured. If this is to much for you, and you say, "oh what a flowery pot I am with such a proposal," then think about "the concept" being used.

The struggle for the emotive language to be explained to the everyday person, as if, the Synesthesist was being simple in their explanation? A "one inch" equation perhaps? They should be so lucky that they could explain themself while they toy with the world and try and make sense of it. That is how different it can be in finding some result of clarification.

That is how foreign I would lead you to believe, that if I wish to communicate, that any language developed, was speaking directly to the source of all expressions, as if they had a geometrical explanation to it. Use of Riemann is understood i this way. It did not divorce him from his teacher, but added vitality tthe way in which we seen Gaussian Arcs and all.

The Magic Square

Hans_Freudenthal

Hans Freudenthal (September 17, 1905 – October 13, 1990) was a Dutch mathematician born in Luckenwalde in Germany into a Jewish family. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education.


I had to think sometimes that what was common knowledge can sometimes be wrapped in up the language we use. So imagine for a time that you will go out and change the way we see the world and add this particular model of String theory just to confuse the heck out of us all.

Lincos

Lincos (an abbreviation of the Latin phrase lingua cosmica) is an artificial language first described in 1960 by Dr. Hans Freudenthal and described in his book Lincos: Design of a Language for Cosmic Intercourse, Part 1. It is a language designed to be understandable by any possible intelligent extraterrestrial life form, for use in interstellar radio transmissions.


Do you want to take the time and consult with the aliens we have on this earth? :) Now surely you know I jest, because of the way in which this model asks a us to look at the world. What use you say?

Please don't confuse this language adaptation to the "ignorance and arrogance" of the "Lincos," a being something other then the human beings who are trying to get a GRIP ON OUR PERSPECTIVES. ASKING US TO SEE IN WAY THAT WE ARE NOT TO ACCUSTOM Too.

Were it Perfect, Would it Work Better?-Bruno Bassi

5.1. Communication vs Formalization

The idea of applying achievements from symbolic logic to the design of a complete language is deeply linked to a strong criticism towards the dominant 20th century trend of considering formal languages as a subject matter in themselves and of using them almost exclusively for inquiries about the foundations of mathematics. "In spite of Peano's original idea, logistical language has never been used as a means of communication ... The bounds with reality were cut. It was held that language should be treated and handled as if its expressions were meaningless. Thanks to a reinterpretation, 'meaning' became an intrinsic linguistic relation, not an extrinsic one that could link language to reality" (p. 12).

In order to rescue the original intent of formal languages, Lincos is bound to be a language whose purpose is to work as a medium of communication between people, rather than serve as a formal instrument for computing. It should allow anything to be said, nonsense included. In Lincos, "we cannot decide in a mechanical way or on purely syntactic grounds whether certain expressions are meaningful or not. But this is no disadvantage. Lincos has been designed for the purpose of being used by people who know what they say, and who endeavor to utter meaningful speech" (p. 71).

As a consequence, Lincos as a language is intentionally far from being fully formalized, and it has to be that way in order to work as a communication tool. It looks as though the two issues of communication and formalization radically tend to exclude each other. What Lincos seems to tell us is that formalization in the structure of a language can hardly generate straightforward understanding.

Our Dr. Freudenthal saw very well this point. "there are different levels of formalization and ... in every single case you have to adopt the one that is most adaptable to the particular communication problem; if there is no communication problem, if nothing has to be communicated in the language, you can choose full formalization" (Freudenthal 1974:1039).

But then, how can the solution of a specific communication problem ever bring us closer to the universal resolution of them all? Even in case the Lincos language should effectively work with ETs, how could it be considered as a step towards the design of a characteristica universalis? Maybe Dr. Freudenthal felt that his project needed some philosophical justification. But why bother Leibniz?

Lincos is there. In spite of its somewhat ephimeral 'cosmic intercourse' purpose it remains a fascinating linguistic and educational construction, deserving existence as another Toy of Man's Designing.

Saturday, January 06, 2007

Mersenne Prime: One < the Power of two


It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.)


This is part of the education of my learning to understand the implications of the work of Riemann in context of the Riemann Hypothesis. Part of understanding what this application can do in terms helping us to see what has developed "from abstractions of mathematics," to have us now engaged in the "real world" of computation.

In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two multiplied by itself a certain number of times. Note that one is a power (the zeroth power) of two. Written in binary, a power of two always has the form 10000...0, just like a power of ten in the decimal system.

Because two is the base of the binary system, powers of two are important to computer science. Specifically, two to the power of n is the number of ways the bits in a binary integer of length n can be arranged, and thus numbers that are one less than a power of two denote the upper bounds of integers in binary computers (one less because 0, not 1, is used as the lower bound). As a consequence, numbers of this form show up frequently in computer software. As an example, a video game running on an 8-bit system, might limit the score or the number of items the player can hold to 255 — the result of a byte, which is 8 bits long, being used to store the number, giving a maximum value of 28−1 = 255.


I look forward to the help in terms of learning to understand this "ability of the mind" to envision the dynamical nature of the abstract. To help us develop, "the models of physics" in our thinking. To learn, about what is natural in our world, and the "mathematical patterns" that lie underneath them.

What use the mind's attempt to see mathematics in such models?

"Brane world thinking" that has a basis in Ramanujan modular forms, as a depiction of those brane surface workings? That such a diversion would "force the mind" into other "abstract realms" to ask, "what curvatures could do" in terms of a "negative expressive" state in that abstract world.

Are our minds forced to cope with the "quantum dynamical world of cosmology" while we think about what was plain in Einstein's world of GR, while we witness the large scale "curvature parameters" being demonstrated for us, on such gravitational look to the cosmological scale.

Mersenne Prime


Marin Mersenne, 1588 - 1648


In mathematics, a Mersenne number is a number that is one less than a power of two.

Mn = 2n − 1.
A Mersenne prime is a Mersenne number that is a prime number. It is necessary for n to be prime for 2n − 1 to be prime, but the converse is not true. Many mathematicians prefer the definition that n has to be a prime number.

For example, 31 = 25 − 1, and 5 is a prime number, so 31 is a Mersenne number; and 31 is also a Mersenne prime because it is a prime number. But the Mersenne number 2047 = 211 − 1 is not a prime because it is divisible by 89 and 23. And 24 -1 = 15 can be shown to be composite because 4 is not prime.

Throughout modern times, the largest known prime number has very often been a Mersenne prime. Most sources restrict the term Mersenne number to where n is prime, as all Mersenne primes must be of this form as seen below.

Mersenne primes have a close connection to perfect numbers, which are numbers equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. In the 18th century, Leonhard Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist (any that do have to belong to a significant number of special forms).

It is currently unknown whether there is an infinite number of Mersenne primes.

The binary representation of 2n − 1 is n repetitions of the digit 1, making it a base-2 repunit. For example, 25 − 1 = 11111 in binary


So while we have learnt from Ulam's Spiral, that the discussion could lead too a greater comprehension. It is by dialogue, that one can move forward, and that lack of direction seems to hold one's world to limits, not seen and known beyond what's it like apart from the safe and security of home.

Tuesday, January 17, 2006

A New Way of Seeing?

Sorry couldn't resist. How many before us in our speculations and way of seeing? :)

When looking at Gaussian coordinates, the very idea that our views of "length of lines" had to have another way in which to interpret how we would see such divergences in the UV considerations. Now I might use UV differently then most, but it is always in context of gaussian coordinates that I always do this.

If I had created a triangle in much the same context(empheral qualties) we might have seen in 2D idealizations then, in how would you transfer such thinkng to three dimensional view held in context of the spiral, and the ever widening primes? These views would have to be locked in Gaussian interpretations, whether they came from Riemann or not. But such trancendance to 5d worlds had to have interpetation that would make you see this in other ways?

Saccherri introdces us to the 5th postulate and we move accordingly into the views of Riemann and others, in ways that are different.

Observations on the Regularity of Prime Number Distribution Peter Marteinson

Stanislaw Ulam’s (1964: 516) most general observation on his famous spiral, that a “property of the visual brain” allows patterns relating to the characteristics of primes to be discovered, may indeed stimulate the mathematical imagination, and inspire further creative attempts at visual pattern recognition in this area, but his spiral (fig.1), like its derivatives, has yet to be successfully interpreted in terms of possible arithmetic principles that can explain the genesis of the known distribution of prime numbers. Of his spiral he says only that it “appears to exhibit a strongly nonrandom appearance” (Stein et al. 1964). A corollary of this somewhat disappointing observation is that Euler’s pessimistic prognosis has yet to be disproved: “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate” (cited by Ivars Peterson in Science News, 5/4/2002).


While it is never easy to take it all in it seems certain phrases and sentence structure stand out as important. While they may seem familiar I refrain from specify what this is, while I continue the search.

Topological Primes

The combinatorial concept of cartesian product allows decomposition of topological structure. Thus, a cylinder is the cartesian product of the circle and the line segment.

This provides for what might be called "topological primes" -- comparable to prime numbers in arithmetic wich form composite numbers. Thus, every mathematical structure may be considered as formed of the basic components:

vertex;
edge;
face;
hole;
cross-cap.


See Also:

  • Riemann's Hypothesis: A Pure Love of Math
  • Friday, December 16, 2005

    Grue and Bleen

    Brian Greene:
    In the late 1960s a young Italian physicist, named Gabriele Veneziano, was searching for a set of equations that would explain the strong nuclear force, the extremely powerful glue that holds the nucleus of every atom together binding protons to neutrons. As the story goes, he happened on a dusty book on the history of mathematics, and in it he found a 200-year old equation, first written down by a Swiss mathematician, Leonhard Euler. Veneziano was amazed to discover that Euler's equations, long thought to be nothing more than a mathematical curiosity, seemed to describe the strong force.

    He quickly published a paper and was famous ever after for this "accidental" discovery.


    If one did not seek to find a "harmonial balance" where is this, then what potential could have ever been derived from such situations about the possibilties of a negative expression geometriclaly enhanced?

    Because the negative attributes have not added up to much in production of anti matter, have we assigned a conclusion to the world of geometerical propensities to not encourge such things a topological maps?

    The puzzle to the right(above) was invented by Sam Loyd. The object of the puzzle is to re-arrange the tiles so that they are in numerical order.

    The puzzle forms a model of how the positron moves in Dirac's theory. The numbered tiles represent the negative-energy electrons. The hole is the positron. When a negative-energy electron falls into the hole, the hole appears to have moved to another position.


    While it would not have seemed likely, such redrawings of the pictures of Albrecht Dürer, this individual might not have caught my attention. I seen the revision of the painting redone, and what was caught in it. You had to really look, to get this sense.

    Saturday, October 15, 2005

    Lagrange points

    As always, the pictures serve as links, as well as highlighted paragraphs in blue, and having once visited, purple. Pictures and paragraphs that are highlighted in gold are in conjunction and are direct links to sites, as well as fawcetts, within this blog. The neurolgical funcxtion of imagery was designed this way and I would encurgae wikipedia to use this idea in the images that they use. I suspect server updates reduce links back to them, which is retarded since all apragraph staements can be assigned to them quite easily.

    This is the advancement in imagery use that mental powers had to keep pace with in computer developement. We know streaming video is quite useful, so why not the neurological fucntioning of "the image" that your minds can produce, that connect as these highlighted paragraphs can do?



    These ideas make sense when you understand the effects of gravitational variances, and can see, what the effect of a fifth dimensional perspective can do. I think the writer understood what I was saying in article that follows?

    Figure 2 shows a map of the gravity field of the Sun-Earth restricted three body problem. The contours show that the steepest gradients surround the Earth and Sun, with the five Earth Lagrange Points located in equilibrium regions with relatively gentle gradient. L1-L3 are unstable saddle points, and spacecraft positioned here will always drift away from the equilibrium. L4 and L5 are stable equilibria, and objects can orbit here indefinitely. The blue arrows show that L4 and L5 are actually atop a potential hill - it is the additional effect of the "Coriolis force" that makes them stable.
    -----------------------------------------------

    This newly found Interplanetary Superhighway is a perfect example of the overlap between classic analysis and modern numerical techniques. The genius minds of Euler and Lagrange used the new technique of calculus to solve the restricted three body problem and show the existence of these intriguing equilibrium points in space. Now, 200 years later, we are employing our own ground-breaking methods using dynamical systems theory and supercomputers, and taking our first steps along the invisible tunnels stretching through the solar system


    If one didn't understand this application from a fifth dimensional perspective how would "this viewer" made any sense?



    Such develoepments and perspective allow other views to develope in relation to how we see this planet, beyond the bubble enclosures one might have developed and culminates in this Thalean view.:)

    This all leads to the developement of the Thalean view It is mathematically orientated although I have much to learn, I made use of a developing perspective that few would have realized, had they not put these things together. That's what I try to do, anyway.