Showing posts with label Riemann Sylvestor surfaces. Show all posts
Showing posts with label Riemann Sylvestor surfaces. Show all posts

Wednesday, February 07, 2018

Sack's and Ulam Spiral

Numbers on the marked curve are of the form

x2 + x + 41,

the famous prime-generating formula discovered by Euler in 1772.

 Riemann discovered a geometric landscape, the contours of which held the secret to the way primes are distributed through the universe of numbers.

It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.

Sack's spiral-a version of the Ulam spiral

 Sift the Two's and Sift the Three's,
The Sieve of Eratosthenes.
When the multiples sublime,
The numbers that remain are Prime.


Friday, May 24, 2013

Who is the Clockmaker?

Crucifixion (Corpus Hypercubus) - oil painting by Salvador Dalí
I see a clock, but I cannot envision the clockmaker. The human mind is unable to conceive of the four dimensions, so how can it conceive of a God, before whom a thousand years and a thousand dimensions are as one?
  • From Cosmic religion: with other opinions and aphorisms (1931), Albert Einstein, pub. Covici-Friede. Quoted in The Expanded Quotable Einstein, Princeton University Press; 2nd edition (May 30, 2000); Page 208, ISBN 0691070210
The phrase of course stuck in my mind. Who is the clockmaker. I was more at ease with what Einstein quote spoke about with regards to the fourth dimension and here, thoughts of Dali made their way into my head.

The watchmaker analogy, watchmaker fallacy, or watchmaker argument, is a teleological argument. By way of an analogy, the argument states that design implies a designer. The analogy has played a prominent role in natural theology and the "argument from design," where it was used to support arguments for the existence of God and for the intelligent design of the universe.

The most famous statement of the teleological argument using the watchmaker analogy was given by William Paley in his 1802 book. The 1859 publication of Charles Darwin's theory of natural selection put forward an alternative explanation for complexity and adaptation, and so provided a counter-argument to the watchmaker analogy. Richard Dawkins referred to the analogy in his 1986 book The Blind Watchmaker giving his explanation of evolution.

In the United States, starting in the 1960s, creationists revived versions of the argument to dispute the concepts of evolution and natural selection, and there was renewed interest in the watchmaker argument.
I have always shied away from the argument based on the analogy, fallacy and argument, as I wanted to show my thoughts here regardless of what had been transmitted and exposed on an objective level argument. Can I do this without incurring the wrought of a perspective in society and share my own?

I mean even Dali covered the Tesseract by placing Jesus on the cross in a sense Dali was exposing something that such dimensional significance may have been implied as some degree of Einstein's quote above? Of course I speculate but it always being held to some idea of a dimensional constraint that no other words can speak of it other then it's science. Which brings me back to Einstein's quote.

The construction of a hypercube can be imagined the following way:
  • 1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.
  • 2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
  • 3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
  • 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.

So for me it is about what lays at the basis of reality as to question that all our experiences, in some way masks the inevitable design at a deeper level of perceptions so as to say that such a diagram is revealing.

I operate from this principal given the understanding that all experience is part of the diagram of the logic of a visual reasoning in which such examples are dispersed upon our assessments of the day. While Einstein spoke, he had a reason from which such quote espoused the picture he had in his head?

Also too if I were to deal with the subjectivity of our perceptions then how could I ever be clear as I muddy the waters of such straight lines and such with all the pictures of a dream by Pauli?  I ask that however you look at the plainness of the dream expanded by Jung, that one consider the pattern underneath it all.  I provide 2 links below for examination.

This page lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Clicking on any picture will magnify it.

The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.

The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one lower dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.

See Also:

  • Pauli's World Clock

  • Wednesday, March 07, 2012

    Thursday, April 16, 2009

    Sacks Spiral

    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

    Sacks Spiral of prime numbers

    Robert Sacks devised the Sacks spiral, a variant of the Ulam spiral, in 1994. It differs from Ulam's in three ways: it places points on an Archimedean spiral rather than the square spiral used by Ulam, it places zero in the center of the spiral, and it makes a full rotation for each perfect square while the Ulam spiral places two squares per rotation. Certain curves originating from the origin appear to be unusually dense in prime numbers; one such curve, for instance, contains the numbers of the form n2 + n + 41, a famous prime-rich polynomial discovered by Leonhard Euler in 1774. The extent to which the number spiral's curves are predictive of large primes and composites remains unknown.

    A closely related spiral, described by Hahn (2008), places each integer at a distance from the origin equal to its square root, at a unit distance from the previous integer. It also approximates an Archimedean spiral, but it makes less than one rotation for every three squares.

    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.


  • Quantum Mechanics: Determinism at Planck Scale
  • The Whole World is a Stage
  • Nature's Experiment on the Meaning of Weight
  • Thursday, September 11, 2008

    Nature's Experiment on the Meaning of Weight

    "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

    This picture resides at the bottom of my Blogger for a reason. It is to remind me of what was given esoterically for fermentation on what it's meaning may have in my own life. in another culture it might have been referred to as a "Kōan." Persistence, is the key to unlocking these "time capsules." Accessibility, is the rule of thumb about "infinite regressions" under a logical format of reasoned inductive/deductive application. The modelled approach here recognized under the picture of Raphael and the school's of Athens early descendants. What place does this have for any student who has a teacher that resides in them? What will be your accomplishment in life?

    If the heart was free from the impurities of sin, and therefore lighter than the feather, then the dead person could enter the eternal afterlife

    You had to know what pain and suffering is, in relation to sin. Your conscience rules you and in it there is a "voice of reason." Sets the trials and tribulations before you. And somehow our youth have detached themself from any causations as to their destinies?

    If you have never considered what "truth(feather) had for meaning in historical times" how would you know what is being imparted to the youth of today? That what is now seen/felt in the world, is not recognized. Choosing "the heart" in that sense, and a scale to boot, I push forward this relation here under the auspice of "weight and measure as a paradox for examination. That such a measure in scale of those ancient times, might spark one to think about the experiment I am showing under this blog post heading.

    This post has been brewing for some time, and only recently the comment made was included to help ignited some of the thoughts I have had about the "weight of something" tied to some method of sound and a gravitational inclination to the pursuances I have of inclusion to colour and sound respectively.

    I think this is a nice historical consideration toward a "new culture."

    I was looking for something even more natural.

    No "sky hooks" but some place to which this process is tied.

    A gourd of water perhaps, tied to a string and the weight of this gourd adjusted by volume, to embed in the string different tension properties and sound characterizations.

    I do not mean to discredit the young scientist here by connection of this comment, but to attached to what seems relevant to me the natural appearance of a counting system, to a method of "fingerprinting" accord to a natural explanation of curvatures in space.

    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:Mersenne Prime: One < the Power of two

    Pacal's Triangle See:Blaise Pascal

    Why only this one time( a numbered base not less then one ensuing from Pascal's triangle on Fibonacci) and I think to differ, that Riemann(Riemann Hypothesis) saw a unique opportunity here. To allow nature to express itself in "non-euclidean terms," that makes for the naturalness of sound to exemplify this unique quality, and an attachment to "colour of gravity" which is exhibited toward my understandings.

    That I may have gone "one step further" on the basis of my "emotive, mental and spiritual search correlations. A search, with respect toward the qualities of gravity. Here it could have been made from the insight of Einsteins example of a "pretty girl and a hot stove?" That "pain" and "excitement" have different values when assigned to a relational quality close to "body home," versus, it's gravitational freedoms, and fleeting examples, that beauty, and mental states can have in "time relational values."

    See:Dialogos of Eide: A New Culture?

    Sunday, June 08, 2008

    Who said it?

    At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’. Michael Atiyah

    "No Royal Road to Geometry?"

    Click on the Picture

    Are you an observant person? Look at the above picture. Why ask such a question as to, "No Royal Road to Geometry?" This presupposes that a logic is formulated that leads not only one by the "phenomenological values" but by the very principal of logic itself.

    All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

    I don't think I could of made it any easier for one, but to reveal the answer in the quote. Now you must remember how the logic is introduced here, and what came before Euclid. The postulates are self evident in his analysis but, little did he know that there would be a "Royal Road indeed" to geometry that was much more complex and beautiful then the dry implication logic would reveal of itself.

    It's done for a reason and all the geometries had to be leading in this progressive view to demonstrate that a "projective geometry" is the final destination, although, still evolving?

    Eventually it was discovered that the parallel postulate is logically independent of the other postulates, and you get a perfectly consistent system even if you assume that parallel postulate is false. This means that it is possible to assign meanings to the terms "point" and "line" in such a way that they satisfy the first four postulates but not the parallel postulate. These are called non-Euclidean geometries. Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such.

    In this axiomatic approach, projective geometry means any collection of things called "points" and things called "lines" that obey the same first four basic properties that points and lines in a familiar flat plane do, but which, instead of the parallel postulate, satisfy the following opposite property instead:

    The projective axiom: Any two lines intersect (in exactly one point).

    If you are "ever the artist" it is good to know in which direction you will use the sun, in order to demonstrate the shadowing that will go on into your picture. While you might of thought there was everything to know about Plato's cave and it's implication I am telling you indeed that the logic is a formative apparatus concealed in the geometries that are used to explain such questions about, "the shape of space."

    The Material World

    There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.

    The other reason is just that the complete mapping of E8 is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes to store the map of E8. If you were to write it out on paper in 6-point print (that's really small print), you'd need a piece of paper bigger than the island of Manhattan. This thing is huge.

    Polytopes and allotrope are examples to me of "shapes in their formative compulsions" that while very very small in their continuing expression, "below planck length" in our analysis of the world, has an "formative structure" in the case of the allotrope in the material world. The polytopes, as an abstract structure of math thinking about the world. As if in nature's other ways.

    This illustration depicts eight of the allotropes (different molecular configurations) that pure carbon can take:

    a) Diamond
    b) Graphite
    c) Lonsdaleite
    d) Buckminsterfullerene (C60)
    e) C540
    f) C70
    g) Amorphous carbon
    h) single-walled carbon nanotube

    Review of experiments

    Graphite exhibits elastic behaviour and even improves its mechanical strength up to the temperature of about 2500 K. Measured changes in ultrasonic velocity in graphite after high temperature creep shows marked plasticity at temperatures above 2200 K [16]. From the standpoint of thermodynamics, melting is a phase transition of the first kind, with an abrupt enthalpy change constituting the heat of melting. Therefore, any experimental proof of melting is associated with direct recording of the temperature dependence of enthalpy in the neighbourhood of a melting point. Pulsed heating of carbon materials was studied experimentally by transient electrical resistance and arc discharge techniques, in millisecond and microsecond time regime (see, e.g., [17, 18]), and by pulsed laser heating, in microsecond, nanosecond and picosecond time regime (see, e.g., [11, 19, 20]). Both kind of experiments recorded significant changes in the material properties (density, electrical and thermal conductivity, reflectivity, etc. ) within the range 4000-5000 K, interpreted as a phase change to a liquid state. The results of graphite irradiation by lasers suggest [11] that there is at least a small range of temperatures for which liquid carbon can exist at pressure as low as 0.01 GPa. The phase boundaries between graphite and liquid were investigated experimentally and defined fairly well.

    Sean Carroll:But if you peer closely, you will see that the bottom one is the lopsided one — the overall contrast (representing temperature fluctuations) is a bit higher on the left than on the right, while in the untilted image at the top they are (statistically) equal. (The lower image exaggerates the claimed effect in the real universe by a factor of two, just to make it easier to see by eye.)
    See The Lopsided Universe-.

    #36.Plato on Jun 12th, 2008 at 10:17 am


    Thanks again.

    “I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered.”Harold Scott Macdonald (H. S. M.) Coxeter

    Moving to polytopes or allotrope seem to have values in science? Buckminister Fuller and Richard Smalley in terms of allotrope.

    I was looking at Sylvestor surfaces and the Clebsch diagram. Cayley too. These configurations to me were about “surfaces,” and if we were to allot a progression to the “projective geometries” here in relation to higher dimensional thinking, “as the polytope[E8]“(where Coxeter[I meant to apologize for misspelling earlier] drew us to abstraction to the see “higher dimensional relations” toward Plato’s light.)

    As the furthest extent of the Conjecture , how shall we place the dynamics of Sylvestor surfaces and B Fields in relation to the timeline of these geometries? Historically this would seem in order, but under the advancement of thinking in theoretics does it serve a purpose? Going beyond “planck length” what is a person to do?

    Thanks for the clarifications on Lagrange points. This is how I see the WMAP.

    Diagram of the Lagrange Point gravitational forces associated with the Sun-Earth system. WMAP orbits around L2, which is about 1.5 million km from the Earth. Lagrange Points are positions in space where the gravitational forces of a two body system like the Sun and the Earth produce enhanced regions of attraction and repulsion. The forces at L2 tend to keep WMAP aligned on the Sun-Earth axis, but requires course correction to keep the spacecraft from moving toward or away from the Earth.

    Such concentration in the view of Sean’s group of the total WMAP while finding such a concentration would be revealing would it not of this geometrical instance in relation to gravitational gathering or views of the bulk tendency? Another example to show this fascinating elevation to non-euclidean, gravitational lensing, could be seen in this same light.

    Such mapping would be important to the context of “seeing in the whole universe.”

    See:No Royal Road to Geometry
    Allotropes and the Ray of Creation
    Pasquale Del Pezzo and E8 Origination?
    Projective Geometries