#quantum logic

Beyond Descartes - Part 5

Reciprocation, Alternation, Decussation
Imaginary and Complex Numbers

(continued from here)

Previously in this blog a number of attempts have been made to explicate the Taoist number line and contrast it with the Western version of the same.  It is essential to do this and to do it flawlessly,  first because different systems of arithmetic result from the two, and secondly because the mandalic coordinate system is based on the former perspective while the Cartesian coordinate system is based on the latter.^[1]

What has been offered earlier has been accurate to a degree, a good first approximation. Here we intend to present a more definitive account of the Taoist number line,  describing both how it is similar to and how it differs from the  Western number line  used by Descartes in formation of his coordinate system.  This will inevitably transport us  well beyond that comfort zone offered by the more accessible three-dimensional cubic box that has heretofore engaged us.

Both Taoist and Western number lines observe directional locative division of their single dimension into two major partitions:  positive and negative for the West;  yinandyang for Taoism.^[2]  There the similarities essentially end.  From its earliest beginnings Taoism recognized a second directional divisioning in its number line, that of manifest/unmanifestorbeingandbecoming.^[3]  The West never did such.  As a result, some time later the West found it necessary to invent imaginary numbers.^[4][5]

Animaginary number is a complex number that can be written as a real numbermultiplied by theimaginary uniti, which is defined by its property i² = −1. [Wikipedia]

Descartes knew of these numbers but was not particularly fond of them.  It was he, in fact, who first used the term “imaginary” describing them in a derogatory sense. [Wikipedia]  The term “imaginary number” now just denotes a complex number with a real part equal to 0,  that is, a number of the form bi. A complex number where the real part is other than 0 is represented by the form a + bi.

In place of the complex plane, Taoism has (and always has had from time immemorial)  a plane of potentiality.  An explanation of this alternative plane was attempted earlier in this blog,  but it can likely be improved. This post has simply been a broad brushstrokes overview. In the following posts we will look more closely at the specifics involved.^[6]

(continuedhere)

Image (lower): A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. “Re” is the real axis, “Im” is the imaginary axis, and i is the imaginary unit which satisfies i² = −1. Wolfkeeper at English Wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Notes

[1] The arithmetic system derived from the Taoist number line can perhaps best be understood as a  noumenal  one. It applies to the world of ideas rather than to our phenomenal world of the physical senses, but it may also apply to the real world, that is, the real real world which we can never fully access.

Much of modern philosophy has generally been skeptical of the possibility of knowledge independent of the physical senses, and Immanuel Kant gave this point of view its canonical expression: that the noumenal world may exist, but it is completely unknowable to humans. In Kantian philosophy, the unknowable noumenon is often linked to the unknowable “thing-in-itself” (Ding an sich, which could also be rendered as “thing as such” or “thing per se”), although how to characterize the nature of the relationship is a question yet open to some controversy. [Wikipedia]

[2] From the perspective of physics this involves a division into two major quanta of charge, negative and positive, which like yinandyang can be either complementary or opposing.  Like forces repel one another and unlike attract. This is the basis of electromagnetism, one of four forces of nature recognized by modern physics. But it is likely also the basis, though not fully recognized as such, of the strong and weak nuclear forces, possibly of the force of gravity as well. I would suspect that to be the case. The significant differences among the forces  (or force fields, the term physics now prefers to use)  lie mainly, as we shall see, in intricate twistings and turnings through various dimensions or directions that negative and positive charges undergo in particle interactions.

[3] It is this additional axis of probabilistic directional location, along with composite dimensioning, both of which are unique to mandalic geometry, that make it a geometry of spacetime,  in contrast to Descartes’ geometry which, in and of itself, is one of space alone. The inherent spatiotemporal dynamism that is characteristic of  mandalic coordinates  makes them altogether more relevant for descriptions of particle interactions than Cartesian coordinates, which often demand complicated external mathematical mechanisms to sufficiently enliven them to play even a partial descriptive role, however inadequate.

[4] In addition to their use in mathematics, complex numbers, once thought to be  "fictitious" and useless,  have found practical applications in many fields, including chemistry, biology, electrical engineering, statistics, economics,  and, most importantly perhaps, physics..

[5] The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them “fictitious” during his attempts to find solutions to cubic equations in the 16th century.  At the time, such numbers were poorly understood,  consequently regarded by many as fictitious or useless as negative numbers and zero once were. Many other mathematicians were slow to adopt use of imaginary numbers, including Descartes, who referred to them in his La Géométrie, in which he introduced the term imaginary,  that was intended to be derogatory. Imaginary numbers were not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).  Geometric interpretation of  complex numbers as points in a complex plane  was first stated by mathematician and cartographer Caspar Wessel in 1799. [Wikipedia]

[6] What I have called here the plane of potentiality occurs only implicitly in the Taoist I Ching but is fully developed in mandalic geometry. It may be related to  bicomplex numbers  or tessarines in abstract algebra, the existence of which I only just discovered. Unlike the quaternions first described by Hamilton in 1843, which extended the complex plane to three dimensions, but unfortunately are not commutative,  tesserines or bicomplex numbers  are hypercomplex numbers in a commutative,  associative  algebra over real numbers,  with two imaginary units (designated i and k). Reading further, I find the following fascinating remark,

The tessarines are now best known for their subalgebra of real tessarines t = w + y j, also called split-complex numbers, which express the parametrization of the unit hyperbola. [Wikipedia]

The rectangular hyperbola x2-y2 and its conjugate, having the same asymptotes. The Unit Hyperbola is blue, its conjugate is green, and the asymptotes are red. By Own work (Based on File:Drini-conjugatehyperbolas.png) [CC BY-SA 2.5],via Wikimedia Commons

Note to self:  Also investigate Cayley–Dickson constructionandzero divisor. Remember,  this is a work still in progress,  and if a  bona fide mathematician  believes division by zero is possible in some circumstances,  (as is avowed by mandalic geometry), I want to find out more about it.

© 2015 Martin Hauser

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Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
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Beyond Descartes - Part 4
Directional Locatives

(continued from here)

Descartes derives his directional locatives from considerations of human anatomy, as does most of Western culture. The descriptive terms generally used for orientation purposes include left/right;up/down; and forward/backward.^[1] The first two sets have been extended also to refer to the cardinal directions, North/South and East/West.

To the degree that they conform to Cartesian coordinates, mandalic coordinates adhere to this schema as well.  However, mandalic geometry and the Taoist I Ching upon which it is largely based constitute a system of combinatorial relationships that is rooted mainly in  radial symmetry rather than bilateral symmetry. For mandalic coordinates, the principal directional locatives can be characterized as  divergentandconvergent, and the principal movements or changes in position, as centrifugalandcentripetal.^[2]

One of the important consequences of this alternative geometric perspective is that the frame of reference as well as the complex pattern produced are more integrative than in the method of Descartes. Looked at another way, Descartes is most enamored by specification of location of individual points whereas mandalic geometry is more concerned with relationships of parts - and the overall unification of the entire complex holistic system.^[3]

From this one seemingly small difference an enormous disparity grows in a manner reminiscent of chaos theory.^[4] Cartesian coordinates and mandalic coordinates can be made commensurate, but remain after all two exclusive systems of spatial awareness,  leading to very disparate results arising out of what seem small initial differences.^[5]

(continuedhere)

Image (bottom): Animation of a double compound pendulum showing chaotic behaviour. By Catslash (Own work). [Public domain], via Wikimedia Commons.^[6]

Notes

[1] Such terminology is of little use, despite its biological origins, to an amoeba or octopus,  not to mention those  extraterrestrials  who have been blessed with a second set of eyes at the back of their heads. (We wuz cheated.)

[2] To be more correct, the radial symmetry involved is of a special type. It is not simple planar radial symmetry, nor even the three-dimensional symmetry of a cube and its circumscribed and inscribed spheres. It is all of those but also the symmetry involved in all the different faces of a six-dimensional hypercube and the many relationships among them.

[3] To be fair, Descartes eventually gets around to relating his points in a systematic whole we now know as analytic geometry (1,2).  But as great an achievement though it might be,  Cartesian geometry  lacks the overarching cosmographical implications which characterize mandalic geometry and the I Ching. Descartes’ system is purposed differently, arising as it does out of a very different world view. To paraphrase George Orwell,

“All geometries are sacred, but some geometries are more sacred than others.”

[4] Chaos theory was summarizedbyEdward Lorenzas:

“When the present determines the future, but the approximate present does not approximately determine the future.”

[5] An example of one unique result of mandalic coordination of space is the generation of a geometric/logical probability wave of all combinatorial elements that occur in the 6D/3D hybrid composite dimension specification of the system. I envision this as offering a possible model at least,  if not an actual explanation, of the  probabilistic nature  of quantum mechanics.  Extrapolating this thought to its uttermost conclusion, it is not entirely inconceivable, to my mind at least, that probability itself might be the result of composite dimensioning. (And for such a brash remark I would almost surely be excommunicated from the fold were I but a member.)

[6] Starting the pendulum from a slightly different initial condition would result in a completely different trajectory.  The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. [Wikipedia]

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
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Beyond Descartes - Part 3
Logic Gates and Switches: Introduction

(continued from here)

It has been often noted throughout this work that mandalic geometry does not view points as fundamental geometrical elements in the manner Descartes and Euclid do. It considers them to be evanescent communions of two or more dimensions.  This  alternative perspective  conveys further the insight  that such conjoint formative interface locations both separate and connect. They are both boundaries and tipping points between all the participating dimensions,  what I have whimsically referred to  previously as dimension interchange lanes.  This is a far cry from the way Descartes regards and handles hispoints.

Descartes’points are locations, pure and simple, defining occupants of a uniform geometrical space. They don’t really doorattempt anything; they simply are.  They do not act,  but are acted upon by the equations of Cartesian geometry.  The  points themselves,  for all the reality Descartes attempts to imbue them with, turn out,  when the curtain is drawn,  to be no more capable of mustering an original thought  than is  the Scarecrow in  L. Frank Baum’s  The Wonderful Wizard of Oz.  Being of feeble mind themselves,  they just sit there awaiting brainy algebra to act upon them. In and of themselves,  beyond determining location,  they are essentially impotent.^[1]

A useful way to apprehendpoint locations of mandalic coordinates is to  interpret them  as  logic gates  which can handle  transition operations in a variety of different ways  depending upon the  dimension amplitudes verged on.  Passage through such locations is potentially bidirectional,  in theory if not always in actuality at a given moment, so they accommodate both  convergent and divergent flows  throughout varied amplitude levels of the mandalic structure.  To wit,  they can promote both  differentiationandpotentialization  phases of an evolving process.  Because these points arise through confluence of dimensions,  they bear within their transitory being information imparted by the participating dimensions.  Contrary to Descartes’ simpleminded points, these points have the capacity to encode an intelligence derived from their parent dimensions.^[2]

In electrical engineering,aswitch is an electrical component that can control an electrical circuit  by initiating or interrupting the current  or by diverting it from one conductor to another.  The most usual configuration consists of  a manually operated electromechanical device  having  one or more sets of electrical contacts.  These contacts are connected to external circuits. Each set of contacts can be in either of two states: either “closed” meaning the contacts are touching and electricity can flow between them, or “open”, meaning the contacts are separated in which case the switch is nonconducting. The mechanism that brings about the transition between these two states - openorclosed - can be either a “toggle”  (flip switch for continuous “on” or “off”)  or  “momentary”  (depress and hold for “on” or “off”) type.

Understand that logic gates don’t apply only to electronic devices nor are they controlled only by such devices. The concepts and methodologies involved go far beyond simple electronics.

• Logic gates are primarily implemented using diodes or transistors acting as electronic switches, but can also be constructed using vacuum tubes, electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathematics that can be described with Boolean logic. Wikipedia

For our purposes here and now, we need only mention that scalar numbers and vectors can be implemented in the context of Boolean logic as well.  Indeed, the incessant complex cotillion performed by subatomic particles can likely be subjected to such an analysis or one similar.^[3] And, of course, also digital circuits and computer architecture.

This has been just an introductory teaser to the topic of logic gates in mandalic geometry.  I’m getting my feet wet now myself. This is all still quite new to me so we’ve barely scratched the surface here.  An upcoming post will survey the logic gates and switches identifiable among groups of transliteration Cartesian coordinates and mandalic coordinates. This may take a while to materialize, but I think will be worth the wait.  And in case I forget to bring up the subject of how fractals fit into all this sometime in the next month or two, remind me please that I intended to.

(continuedhere)

Notes

[1] This could be a mathematician’s beautiful dream, but a physicist’s abhorrent nightmare.

[2] Although this statement pertains especially to composite dimension points, it is true, to a degree, of ordinary three-dimensional points as well when viewed in a manner similar to that using trigram tranliterations of Cartesian triads.  This means then that Cartesian coordinates could do the same and to the same degree, if  they were handled in the same manner as trigram coordinates are. The point is they are not and presumably never were.

[3] With that last remark I likely committed quantum mechanical heresy. If I in fact did, so be it. If it doesn’t quite hit the intended mark we can refer to it as steampunk mechanics.

Image (lower): Boolean lattice of subsets. KSmrq. Licensed under CC BY-SA 3.0viaCommons.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x + 1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
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Earlier to Later Heaven: Fugue VII Beyond Descartes - Part 2
A Different Zero

(continued from here)

Mandalic geometry has been formulated in such a manner as to be fully commensurate with Descartes’ coordinate system. Firstly, because it can be.  Beyond that,  because Descartes’ system is known throughout the world, and is endorsed by all conversant in disparate fields of science and mathematics. Moreover, the Cartesian coordinate system is a special case of the mandalic coordinate system,  bearing a relationship to it analogous to that which Newtonian mechanics does to quantum mechanics.

One of the fundamental differences lies in the way the two regard zero locations. Descartes, taking his cue from the Western number line, constructs a coordinate system which envisages a single common origin to all three dimensions, while maintaining between those dimensions a rigid uncompromising distinction. Mandalic geometry views dimension as primary rather than points, lines, or two or three dimensional figures. It does not regard dimensions as intrinsically separate in the manner in which they  exist and relate  to one another.  This allows for a far greater degree of flexibility of what we view as parts of the system, including the possibility of folding each into another,  through different dimensions as well as the same dimension.

For Descartes, zero is the empty location, the no man’s land where positive and negative vectors of each dimension invert or fail to invert.  A negative vector acting on a positive vector or another negative vector will cause inversion.  A positive vector, acting on a negative vector or another positive vector, will not. For mandalic geometry, zeros are that, but more. They are dimension interchange lanes,  and also locations of dimensional amplitudetransition.^[1]

Descartes, influenced still by the number line, proceeds to build a geometric universe based largely on scale. It is an imposing edifice nearly purely divergent,  constructed from three largely independent linear axes of evolutionary zeal.  Taoist cosmology and mandalic coordinates equally eschew an impressive but mundane number line in pursuance of complex twisting and intertwining of parts evolved on the underlying principles of modularity, repetition, reflection, relationship and recursion.^[2]

These are two very different universes of logic.  Descartes’ approach leads to a description of space as being homogenous, isotropic, and fixed while that of mandalic geometry leads alternatively to a spacetime which is inhomogeneous, anisotropic and dynamically variable.^[3] For Descartes space is a background arena,  the theater in which all events transpire.^[4] For mandalic geometry,  space-time is foreground and background both. It is the sole ground which defines the nature of reality.

(continuedhere)

Notes

[1] The first,  dimensional interchanges,  occur in the Cartesian coordinate system but are generally neither recognized nor treated as such. Dimensional amplitude transition locations do not occur in Cartesian coordinates,  nor are they found in the simple 3D trigram Cartesian equivalent,  reproduced in the upper diagram above, as they are a manifestation only of compositing of two or more dimensions. They are attributes of all hybrid composite dimensional systems,  for our purposes here, either the 6D/3D hybrid mandalic system of hexagrams,  the 4D/2D hybrid mandalic system of tetragrams,  or the 2D/1D hybrid mandalic system of bigrams.

[2] An important consequence here is that Descartes’ number line-based axes each contain a single zero. When mandalic coordinates are scaled up beyond the basic modular unit, every even number maintains all characteristics of the initial zero, including, most significantly, its multipotentiality. This is a basic axiomatic result of the intermingling, sharing nature of mandalic structure.

[3] It is this variability and dynamism of mandalic coordinates that make the method potentially suitable to mappings of subatomic particles as these are similarly variable and dynamic,  sharing importantly also the ability of exchanges / interchanges among their diverse numbers.

[4] Witness for example how Descartes exploits his newly formed coordinate system to stage, what was then, a cutting-edge geometric exposition of algebra, now referred to as analytic geometry. Mandalic geometry employs coordinates which are pre-invested with the ability to directly impart information regarding spatial transmutations themselves, without requirement of any intermediary.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x + 1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

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Earlier to Later Heaven: Fugue VI Beyond Descartes - Part 1

(continued from here)

In this post we take a short detour within our current central topic, that of relationship of Earlier Heaven and Later Heaven arrangements of the trigrams. The new material included here grew out of ruminations on the aforesaid primary topic though,  and is actually not so much a detour as a preparing the way for what I hope will be the eventual solution of our problem at hand.

Mandalic geometry, as we’ve seen, is fully commensurate with the coordinate system of Descartes, but its principal forebears lie elsewhere. It is derived largely from Taoist and pre-Taoist thought structures, most importantly the I Ching,  the earliest strata of which were formed before the separation of rational and irrational thought in the history of human cognition. As a result it is capable of far exceeding the possibilities of the Cartesian coordinate system, a product of the Enlightenment and Age of Rationalism. It offers geometry the possibility of a structural fluidity and a functional variability that Cartesian geometry lacks.^[1]

From the very beginning of this project I’ve been much puzzled by the lack in traditional Chinese thought  of a symbol corresponding to the zero of the Western number line and number theory.^[2] Traditional Asian thought does not uniformly lack a zero symbol.^[3] And yet the I Ching and Taoism manage well enough without one, electing to base their numerical relationships instead entirely on combinatorics involving permutations of yinandyang – what we in the West call  negativeandpositive – through multiple dimensions. It is an entirely different perspective arising out of a very different worldview.^[4]

What Taoism invented in the process was a unique,  thoroughly self-consistent brilliant system of logic/geometry/combinatorics which has been masquerading, all these many centuries,  as “just a method of divination.”^[5]  In essence, Chinese thought invented a discrete number system and geometry, one based on vectors rather than scalars, a vector geometry that can be extrapolated to any desired number of dimensions. The I Ching settles for just six,  the first whole number multiple of three. That is complicated enough.^[6][7]

(continuedhere)

Notes

[1] For one example of the advantages such variability and fluidity offer, in this particular case in creating  dynamic,  phase-shifting forms of nanomaterials,  see here.

[2] For a short history of the concept of zeroseeWho Invented Zero?

[3] The West, after all, derived its zero symbol ultimately via India.

[4] One might well speculate whether the significant root difference in world view between traditional Indian and Chinese thought lay in the fact that Indian mathematicians could have created a Zero out of nothingness (Śūnyatā),  a key term in Mahayana Buddhism and also some schools of Hindu philosophy while Taoist thought did not include a concept of nothingness. Instead it conceived of a formlessness prior to manifestation. In Taoist cosmology Taiji is a term for the “Supreme Ultimate” state of undifferentiated absolute and infinite potential,  the oneness before duality,  from which  yinandyang  originate.  So it might be that lacking a concept of nothingness forestalled invention of a zero symbol.  Still, it also allowed creation of an original,  unique holistic philosophy of reality, found perhaps nowhere else.

[5] The Russian philosopher, mathematician and authorPeter D. Ouspensky (1878-1947)  relates an apocryphal legend regarding the origin of the Tarot,  the moral of which has significance also to the history of the I Ching.

[6] In its emphasis on vector analysis and primacy of dimension the philosophy which underlies the I Ching and mandalic geometry  shares some characteristics of Clifford algebra.

[7] One of the important things with respect to physics I hope to show with mandalic geometry is that it is possible to construct an integrated geometrical / logical system which is self-sufficient and self-consistent, capable of modeling interactions of subatomic particles of the Standard Model and then some.  This goal is,  I believe,  approximated in mandalic geometry by meticulous coupling of the methodologies of composite dimension and trigram toggling,  although it quickly becomes apparent that a system based upon what is after all a relatively small number of dimensions - six in the case in point - becomes vastly complex and difficult to follow, at least initially.  One can’t help wondering how physics will be able to correlate all the intricate data resulting from its countless particle accelerator collisions and combine it into a consistent whole without some very fancy mental acrobatics on the part of theoretical physicists.  Without a suitable logical scaffold that might take an inordinately long time to achieve.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x + 1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

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