Category Archives: Space and Time

Four-Dimensional Vistas

The Fourth Dimension has been an interest of mine since I was a child. I’m not sure when I first heard about it, but I still have my coverless copy of “Geometry of 4 Dimensions” by Henry Parker Manning that I bought in a used book store. (I wonder why it hasn’t ever been reissued by Dover?) Maybe I heard about the fourth dimension in some science fiction TV movie, or in some mathematical survey book like “Mathematical Snapshots” or “Mathematics and the Imagination”.

Once I tried to explain to my best friend about my newly discovered insight how a hypercube could be folded up in four-dimensional space from its so-called three-dimensional net consisting of eight cubes, just as a regular three-dimensional cube could be folded up from its two-dimensional net of six squares. This 3D net somewhat resembling a cross is famously seen in Dali’s “Crucifixion (Corpus Hypercubus),” although I probably didn’t refer to this painting in my explanation.

I’m not sure who came up with the take-away message from my exposition, but it remains clear in my memory that the “junk in the middle” of the hypercube was a piece of the fourth dimension, just as the faces of a cube enclose a piece of our normal third dimension.

I recently came across Claude Fayette Bragdon, architect, author, draughtsman, stage designer, and mystic. At first I was interested in his drawings found on-line. His book “Four-Dimensional Vistas” started off with a good if overly wordy introduction to the concept of the fourth dimension. But then he suggests that many esoteric concepts such as the meaning of dreams, reincarnation, past-life regression, prognostication, ESP, etc. could possibly be explained by higher dimensional space or even higher dimensional time.

Even though I initially found many of these hypotheses too far-fetched for my tastes, I still found some interesting ideas to mull over in this little book.

Further Reading:

https://en.wikipedia.org/wiki/Claude_Fayette_Bragdon

https://www.gutenberg.org/ebooks/author/4128

https://theosophy.wiki/en/Claude_Fayette_Bragdon

https://theosophyart.org/2018/03/30/claude-fayette-bragdon/

https://rbscp.lib.rochester.edu/bragdon-family-papers-claude-bragdon-architectural-drawings

The art of Claude Fayette Bragdon, 1866–1946

https://rbscp.lib.rochester.edu/3514

https://bauarchitecture.com/research.loshuworldofwonderous.shtml

Claude Fayette Bragdon / Four-Dimensional Vistas (1930)

Claude Fayette Bragdon / The Beautiful Necessity (1910)

Claude Fayette Bragdon / Architecture and Democracy (1918)

For my gratuitous anime tie-in, Bragdon’s world-view suddenly reminds me of the anime character  Haruhi Suzumiya, who wished for her aliens, time-travelers, and ESPers so much that she willed them into being. If only she had known about the fourth dimension!

https://reelrundown.com/animation/Anime-Philosophy-1-Melancholy-of-Haruhi-Suzumiya

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The Wolfram Physics Project

When I first started looking at Stephen Wolfram’s latest proposal to solve physics, I was somewhat disappointed. I was rather fond of his previous “New Kind of Science” based on the structural rigidity of cellular automata. However, I am now intrigued by his latest ideas, based on the looser but more flexible basis of networks.

And once you have pithy statements with space, time, energy, and matter (as momenta), you catch my attention:

  • Energy is flux of causal edges
  • through Spacelike hypersurfaces
  • Momentum is flux of causal edges
  • through Timelike hypersurfaces

I confess I haven’t read much about the project yet, but it seems to be using rewriting rules, perhaps similar to the notion of rewriting in Wolfram’s previous framework, cellular automata. Of course, cellular automata and also rewriting rule systems can be computationally universal or Turing complete.

Another idea might be to try some sort of computational metaphysics between nodes like the pi-calculus (or some other process calculus). After all, you have to support quantum entanglement! However if you can encode everything with simpler structures then do it!

Further Reading:

https://www.wolframphysics.org/

https://www.wired.com/story/stephen-wolfram-invites-you-to-solve-physics/

https://writings.stephenwolfram.com/2020/04/how-we-got-here-the-backstory-of-the-wolfram-physics-project/

https://en.wikipedia.org/wiki/Digital_physics

https://www.scientificamerican.com/article/physicists-criticize-stephen-wolframs-theory-of-everything/

https://turingchurch.net/computational-irreducibility-in-wolframs-digital-physics-and-free-will-e413e496eb0a

View at Medium.com

Cellular automata:

https://en.wikipedia.org/wiki/Cellular_automaton

Note this quote for future reference:

The primary classifications of cellular automata, as outlined by Wolfram, are numbered one to four. They are, in order, automata in which patterns generally stabilize into homogeneity, automata in which patterns evolve into mostly stable or oscillating structures, automata in which patterns evolve in a seemingly chaotic fashion, and automata in which patterns become extremely complex and may last for a long time, with stable local structures. This last class are thought to be computationally universal, or capable of simulating a Turing machine.

Rewriting:

https://en.wikipedia.org/wiki/Rewriting

https://en.wikipedia.org/wiki/Semi-Thue_system

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Noether-Pauli-Jung, V2

What happens when the fourfold of Noether’s Theorem is spliced together with the fourfold of Pauli-Jung? Both have Space-Time and Matter-Energy. The former has Conservation and Symmetry, and the latter has Causality and Synchronicity. And if Space-Time and Matter-Energy are both divided into Space and Time and Matter and Energy, one obtains this eight-fold.

Causality means that some action or cause in time (say a process) of things in space can have an effect (another process, say) on different things in space, and Synchronicity means that different events (say processes) separated in space can have non-causal relationships between them.  Conservation means the consistency of a quantity of matter or energy or matter-energy through time, and Symmetry means the consistency of a measure of a structure or form through space.

I am reminded of my fourfold Four Bindings, consisting of Chains, Grids, Blocks, and Cycles. Causality and Synchronicity are Chains (or non-chains for the latter) Space and Time are Grids (or flexible meshes), Matter and Energy are Blocks (or chunks of stuff), and Symmetry or Conservation are Cycles (of the group-theoretic kind or the equivalence class kind or just loops).

Further Reading:

https://en.wikipedia.org/wiki/Causality

https://en.wikipedia.org/wiki/Synchronicity

https://en.wikipedia.org/wiki/Symmetry

https://equivalentexchange.wordpress.com/2012/05/04/noethers-theorem/

https://equivalentexchange.wordpress.com/2018/01/23/atom-and-archetype/

https://equivalentexchange.blog/2016/04/06/four-bindings/

This is a reworking of a previous six-fold diagram that I believe is served better as an eight-fold.

https://equivalentexchange.blog/2018/03/29/noether-pauli-jung/

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Four Forms Make a Universe

How could I not love a paper with this title? I’ve struggled with it for a bit, and I’ve only managed a couple of diagrams relating the author’s LICO (Linear Iconic) alphabet made up of 16 letters. However, I see that there are a few other papers by Schmeikal available on ResearchGate that look easier to understand. But also however, the first one says to read the “Four Forms” paper first!

At any rate, I present a sixteen-fold of the LICO alphabet, and another of the binary Boolean operators that are in a one-to-one mapping with LICO. There is much to understand from these papers, including much syncretism between various mathematical sixteen-folds, so please forgive me if I don’t explain it all with immediate ease. However, I believe it is well worth the effort to understand.

(Please note that the characters of the LICO alphabet are oriented so that the bottoms of the letters are downward, but the Boolean operators are oriented so that the bottoms of the equations are towards the right angles of the triangles.)

The title comes from the result that four elements of LICO can reproduce the other twelve via linear combinations. These four forms are 1) Boolean True (A or ~A), 2) A, 3) B, and 4) A=B. These are within the interior right-hand triangles in the LICO diagram. Of course, it is well known from Computer Science that the NAND operator (~A or ~B) can also generate all other fifteen operators, but this is by multiple nested operations instead of simple Boolean arithmetic. There are several other “universal” binary gates that can do this as well.

Two other representations that have four elements that can generate the other twelve via linear combinations come from CL(3,1), the Minkowski algebra. These representations are called “Idempotents” and  “Colorspace vectors”. Because of this algebra’s association with space and time in relativity, Schmeikal claims that LICO has ramifications in many far-ranging conceptualizations.

Further Reading:

Bernd Schmeikal / Four Forms Make a Universe, in Advances in Applied Clifford Algebras (2015), Springer Basel (DOI 10.1007/s00006-015-0551-z)

https://link.springer.com/article/10.1007/s00006-015-0551-z

At http://www.researchgate.net:

Bernd Schmeikal / Free Linear Iconic Calculus – AlgLog Part 1: Adjunction, Disconfirmation and Multiplication Tables

Bernd Schmeikal / LICO a Reflexive Domain in Space-time (AlgLog Part 3)

https://en.wikipedia.org/wiki/Spacetime_algebra

https://www.allaboutcircuits.com/technical-articles/universal-logic-gates/

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An Eightfold Metaphysics

If one juxtaposes the fourfold Space-Time-Matter-Energy with the fourfold Structure-Function-Part-Action, one sees that there are several relations between them. These are simple common-sense relations, not modern-physics types of relations. Below they are listed by Space-type relations and Time-type relations.

Space is required for the extension of Structures.
Matter constitutes Structures.
Parts in an arrangement make up Structures.

Structures extend and are organized in Space.
Matter is located in Space.
Parts occupy Space.

Time allows the expression of Functions.
Energy is required for the operation of Functions.
Actions in a sequence constitute Functions.

Functions have duration and reoccur in Time.
Energy requires and is dependent on Time.
Actions take place within Time.

Several analogies are also evident in this diagram.

  1. Space : Structures :: Time : Functions
  2. Space : Parts :: Time : Actions
  3. Space : Matter :: Time : Energy
  4. Structures : Parts :: Functions : Actions
  5. Structures : Matter :: Functions : Energy
  6. Parts : Matter :: Actions : Energy

Notice that analogies 1.-3. are “contains” relations, and 4.-6. are “part of” relations. One obtains the nesting of entities:

Space > Structures > Parts > Matter
Time > Functions > Actions > Energy

I suppose that any analogy could be shown in a figure like the one on the right (or even written as A / B // C / D) or with the two squares side by side, and that sets of four or six analogies that overlap could be nicely shown as above. If so, I wonder what can be gained by such representations? Probably individually not so much but with overlaps I think it would be interesting.

Further Reading:

The study of parts and wholes relations is called Mereology.

https://en.wikipedia.org/wiki/Mereology

https://plato.stanford.edu/entries/mereology/

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The Seasons and the Zodiac

This isn’t a bad little diagram of the four seasons along with the twelve zodiac names and symbols. However, it might be oriented wrong by convention or going clockwise instead of counter-clockwise.  Interestingly, old horoscope charts that show what was in the sky (the positions of the zodiac stars and the eight or nine planets in regards to the twelve “astrological houses”) at the time of a person’s birth were shown using the outside ring of twelve triangles instead of the more familiar circle that is used today. The inner square might be for notes or some nice drawing.

Further Reading:

https://en.wikipedia.org/wiki/Zodiac

https://www.completehoroscope.org/envelope-diagram-horoscope.htm

https://equivalentexchange.blog/2014/10/31/the-four-seasons/

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Fourfold Physicalism

It is not enough for a wise man to study nature and truth, he should dare state truth for the benefit of the few who are willing and able to think. As for the rest, who are voluntarily slaves of prejudice, they can no more attain truth, than frogs can fly.

— From Man a Machine, by Julien Offray de La Mettrie

Further Reading:

https://equivalentexchange.wordpress.com/2012/07/03/structure-function/

https://equivalentexchange.wordpress.com/2013/10/09/things-happen/

https://equivalentexchange.wordpress.com/2014/02/01/relations-all-the-way-down/

https://equivalentexchange.wordpress.com/2015/06/09/four-primary-relations/

https://en.wikipedia.org/wiki/Man_a_Machine

Notes:

Structures are built from parts.
Parts are reductions of structures.
Functions are assembled from actions.
Actions are the constituents of functions.

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Atom and Archetype

A few weeks ago I ran across this nice review of the book Atom and Archetype: the Pauli-Jung letters 1932-1958. This is a collection of letters exchanged between psychiatrist Carl Jung and physicist Wolfgang Pauli over a course of years. Evidently, Pauli was quite the metaphysician and Jung was intrigued by Einstein’s physics of relative space and time. Together in dialectic they argued and struggled to join together the disparate notions of mind and matter.

What mainly caught my eye was a diagram that I’ve slightly altered and shown above. I’ve mainly just replaced energy with matter-energy for two reasons: first because matter and energy are inter-convertible and second because matter conditions space. This results in similarity to the fourfold diagram for Lucretius that I’ve shown before, consisting of Particles, the Void, Falling, and Swerving.

Further Reading:

Atom, Archetype, and the Invention of Synchronicity: How Iconic Psychiatrist Carl Jung and Nobel-Winning Physicist Wolfgang Pauli Bridged Mind and Matter

Carl Jung and Wolfgang Pauli / Atom and Archetype: the Pauli-Jung letters 1932-1958, Princeton University Press; Updated edition (July 21, 2014)

https://www.amazon.com/Atom-Archetype-Pauli-Letters-1932-1958/dp/069116147X/ref=sr_1_1?s=books&ie=UTF8&qid=1516537804&sr=1-1&keywords=atom+and+archetype

https://equivalentexchange.wordpress.com/2011/10/10/lucretius-on-the-nature-of-things/

https://equivalentexchange.wordpress.com/2014/02/17/matter-energy-space-and-time/

https://equivalentexchange.wordpress.com/2015/01/16/wave-particle-duality/

https://equivalentexchange.wordpress.com/2012/05/04/noethers-theorem/

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A Digital Universe, V2

A digital universe – whether 5 kilobytes or the entire Internet – consists of two species of bits: differences in space, and differences in time. Digital computers translate between these two forms of information – structure and sequence – according to definite rules. Bits that are embodied as structure (varying in space, invariant across time) we perceive as memory, and bits that are embodied as sequence (varying in time, invariant across space) we perceive as code. Gates are the intersections where bits span both worlds at the moments of transition from one instant to the next.

— George Dyson, from Turing’s Cathedral

Further Reading:

George Dyson / Turing’s Cathedral: the origins of the digital universe

Notes:

Embodied as Structure, Perceived as Memory

Invariant across Time: ¬ΔT
Varying in Space: ΔS

Embodied as Sequence, Perceived as Code

Varying in Time: ΔT
Invariant across Space: ¬ΔS

https://equivalentexchange.wordpress.com/2012/04/12/a-digital-universe/

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The Paradoxes of Zeno

sq_zeno_paradoxes5“Suppose,” said Zeno, “that Achilles and a tortoise are planning to race”.

Such is the beginning of a famous thought experiment by an ancient philosopher. Since athletic Achilles was much faster than the slow tortoise, he let the tortoise start first. But alas, he could never catch up to it, since every time Achilles made it to where the tortoise had been, the tortoise had moved just a little further ahead. Of course Achilles was faster so he had to pass the tortoise quickly unless it had started near the finish line. So, paradox!

Most of the paradoxes of Zeno were about fractions and entireties of time and space. Can an infinite series of fractions of space add up to a finite entirety of space in a finite entirety of time? Some might say that integral calculus solves these basically mathematical problems, yet others think they point to metaphysical issues as regards to the discreteness and the continuity of time and space.

This fourfold reminds me of my previous fourfold Spacetime which dealt with succession (as parts of time), location (as parts of space), extension (as wholes of space), and duration (as wholes of time). It must have been in the back of my mind.

References:

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

https://equivalentexchange.wordpress.com/2011/09/23/spacetime/

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