Future Math

I drew this to help Andrius, in Vilnius, visualize an important identity, used in calculus.  Going to the next higher power, using a tetrahedron, would be a logical next step.

Princeton is where I ran into Wittgenstein's philosophy, including his Remarks on the Foundations of Mathematics (RFM).  When Ludwig wants to "investigate" a math concept, or any concept, he'll often construct "language games" for that purpose.

He does this with intent, as a sketch artist would, to bring forward specific aspects of play that might have been confusing.  Language gets to be a tangled mess sometimes.  For Wittgenstein, philosophy was a lot about disentangling.

Speaking of disentangling, a theme of my writings has been to separate three meanings of 4D.  For example, check out this blog post regarding 4D vs 4D vs 4D.

I'm seeing the 1900s as a period of "shake out" wherein three principal meanings of 4D settled down, each comfortable in its own groove.  They're not identical and to confuse them is to become entangled in a web of language.

Coxeter.4D is what we call "extended Euclidean geometry" meaning (x, y) addressing of R x R (a plane), and (x, y, z) addressing of R x R x R (volume), is extended to any number of real number lines or dimensions.  Regular Polytopes, by Donald Coxeter, is a major contribution within this lineage.

Einstein.4D introduces time as a space-like dimension in that one's perception of time and motion is a function of one's reference frame relative to others, such that those moving at closer to the speed of light age less quickly than clocks at rest.  What's the same across inertial coordinate systems is a space-time interval in which (x, y, z, t) figure in to the computation as four inputs (hence 4D).

Fuller.4D focuses on the primacy of the tetrahedron in the Kantian sense of introducing concepts we cannot shake, such as "enclosure" (container) along with "concavity" (in the cave) and "convexity" (outside the container).  We're born with the a priori idea of being "in" something and the tetrahedron is a topologically minimal concept of having an inside.

Donald Coxeter himself was at pains to disambiguate between his extended Euclidean 4D and Einstein's relativity theory's 4D.

We might summarize this difference as "the difference between a tesseract (hypercube) and a time machine", on the understanding this is precisely the difference some science fiction writers may deliberately confuse, as a plot-driving device.

Donald writes, on page 119 of Regular Polytopes:

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H.G. Wells in The Time Machine, has led such authors as J. W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.  (Dover Publications, 1973)

Donald's relationship with Fuller, another spinner of 4D, was somewhat inevitable in hindsight, and might be characterized as "stormy at first, smoothing out into friendship".

A first point of contention was around intellectual property, around radomes in particular.  The US military honored a US patent in contracting with Fuller's firms for DEW line radomes, situated in Canada.

Donald's son, a Canadian, was hoping to break in to the radar dome business and came up against not owning the relevant patent.  Donald went ballistic, thinking such a fundamental geometry of nature would really have to be open source.

Fuller's having patented the octet truss, in addition to the geodesic dome, likely didn't improve matters.  Both are found all over the place in naturally occurring physical objects.

The softening in their relationship came with the Montreal Expo '67 geodesic dome, a marvel to behold and huge.  Coxeter eventually agreed to let Fuller dedicate Synergetics to him.  Synergetics is where Fuller's use of 4D is made clear.  Bucky writes:

527.702 Geometers and "schooled" people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states --"one-dimensional," "two-dimensional," and "three-dimensional" -- which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of one another nor independently of all the inherent characteristics of all systems and of all systems' inherent complex of interrelationships with Scenario Universe.

...

527.712 All conceptual consideration is inherently four-dimensional. Thus the primitive is a priori four-dimensional, being always comprised of the four planes of reference of the tetrahedron. There can never be any less than four primitive dimensions. Any one of the stars or point-to-able "points" is a system-ultratunable, tunable, or infratunable but inherently four-dimensional.

Coxeter was also impressed by Fuller's discovery of the cuboctahedron number sequence, 1, 12, 42, 92, 162 and so on, his formula for generating this sequence, and generalizable to other shapes.  Clearly Fuller was doing his own thinking to some high degree.

In computer science we now have the concept of "namespace" which is equivalent to the humanities notion of "context".  For example in Python a namespace may be a module, saved in storage as a Python script.

The meaning given to the various names in a script will become evident at runtime.  Likewise in the humanities we appreciate that authors have their own unique ways of using certain words, which both borrow from and contribute to shared usage patterns.

Disentangling the meanings of 4D is important to future math, and to one's level of cultural literacy.  Our Coffee Shops Network has the LCD reveries that help boost one's understanding of these many nuances and subtleties.

by Hollister (Hop) David
(with attribution to me)

Business Model in Bitcoin

Although suggestive of Bitcoin in particular, the CSN model operates with any currency, crypto or not.

Customers of the Coffee Shop buy vended goods (e.g. snacks + coffee), with a percentage of the sale price committing to a Service Pool, which the vendor tallies as after-profit charitable contributions or Good Will (a legit line item in most charts of accounts).

Customers have the option to multiply winnings, drawing from the Service Pool, with said winnings committed to the service organization (or Cause) of the player's choice.  The shop is doubling as a center for engaging in philanthropic behaviors.

Not shown in this diagram:  the customer's option to add victories (donations) to a profile (chronology).  What better way to advertise one's values?  Show a track record of philanthropic giving, thanks to CSN infrastructure.

Other posts spell out more details as to how pooled Good Will might aggregate faster than payout in some phases, slower in others.  Imagine aggregate player ability as a variable, like collective IQ.  Would we want to re-normalize the games i.e. make them harder or easier, to control payout?   Changing the rules punishes those specializing in mastering a given sport.  Changing the payout is easier, though one may argue devaluing a game's relative payout is a change in the rules.

You'll likely notice that Casinos which do feature skill aspects to some games, face similar queries, which is why CSN is sometimes branded a casino chain, even with proceeds going to worthy causes other than self in most transactions, and not "the house" either, as the customers get to earmark and profile accordingly.

CSN shops (nodes) do have some control over the menu however i.e. the range of worthy causes to which winners may commit their winnings.  Customers are free to shop around and/or play from home so shop constraints are not that onerous, merely reflective of a shop's own profile (a way of branding).

4D Meme

Fig 1: three number lines

If three number lines are place tip-to-tip in a zig-zag (non-coplanar), a half-tetrahedron (pink) will be defined, complemented by the other half (black).

Fig 2: three-vector zig-zags
from Fig 110A of Synergetics

Picture the three number lines (each made of two rays), starting as line segments, then growing or subdividing without limit.

These same three number lines, made mutually orthogonal, define the origin (0,0,0) in XYZ.  These number lines likewise extend without limit to define volume or all-space.

Given six number-line edges define volume (3 + 3), we may say space is 3D.  The pink zig-zag of three segments might even feature two right angular turns.  The set {(1,0,0),(0,1,0),(0,0,1)} is cast as the basis set for spanning this space.  Every point is reachable by (x i, y j, z k) where x,y,z are Real numbers.

Alternatively, we may highlight the four vertexes and faces of the full tetrahedron and define res extensa as inherently 4D (self-evident fourness), with quadrays (caltrop coordinates) emanating in four directions from origin (0,0,0,0) and spanning space in linear combinations, no negating operation required.

 

Fig 3: Q-rays

In Fig. 1, R³ is shown as a "4D" tetrahedron (pink and black). In Fig. 3, the four basis vectors (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) with scalar multiplication by non-negative numbers are sufficient to span R³ through linear combinations.

Although we may consider space 4D, given the iconic stature of the tetrahedron, upon subdividing the edges to show growth by subdivision, we obtain a 3rd power sequence relative to the frequency of the edges.  3rd powering is likewise illustrated with a tetrahedron.

Fig 4
Fig. 4: 3rd Powering
from Fig. 990.01 in Synergetics

In the shoptalk of XYZ, we say R³ is 3D.  However, the all-positive basis vectors remain helpless to reach 7/8ths of space without the operation of negation -- flipping 180 degrees -- an operation provided by scalar multiplication in conventional linear algebra, and providing the additional -i, -j, -k of the familiar "jack" pattern.

In the shoptalk of Synergetics, we say the IVM is 4D, given the self-evident fourness of the tetrahedron, yet without denying 1:2:3 power aspect of linear : areal : volumetric growth.  The four all-positive basis vectors do not require the flipping operation of negation (by definition rotation) to span all-space, only vector scaling and vector addition.