Oceans of Numbers 5: right-angles

c = λf c = λv c = λν

The basic relationship between space and time -- wavelength and frequency -- is inherently built into the equation for the speed of light: c = λν. The speed of light, c, is a constant and conserved property -- as is SpaceTime (ST) of which it is intimately connected. ST used in this way is a concept, not an equation.

ST≠ space • time. It is actually space/time=S/T and as Time =1/frequency, S/T=space/(1/ν)=space•ν.

When we let the space (T)=the wavelength (λ), we then have

c = λν = speed of light = space/time = S/T

as the formation of ST and the speed of light are one and the same. A photon is the generation of a quantum unit of ST, both of which are formed at c, the speed of light. Both are conserved, as the Conservation of SpaceTime (CoST) and the constancy of the speed of light.

frequency=cycles/sec=cycles/time -- so ST is space • frequency, or exactly what the speed of light is (c = λν)

That said, what is really being pointed out here is the "right-angle" relationship that exists at the get-go, at the very formation of ST as c.

What is this relationship between space and time (frequency)? Why does it present as a "right-angle" relationship? What is a "right-angle" relationship?

Simple. Multiplication tables are nothing more than simple right angle relationships between the two axis' of a table -- a grid.

The CoST is the big conclusion surrounding the publication of L.U.F. E. in general and the founding of The LUFE Matrix in particular (both in 1985 in print, 2008 online). The latter was a matrix grid formed from the distillation of common SI units into fundamentally more base units of space and time, each forming an axis -- and at right-angles to each other. LightspeedST elaborates on L.U.F.E. (2012) and The LUFE Matrix appearing on its own (print 1991, online 2003).

The long dive into mathematical physics eventually gave way to pure Number Theory mathematics, initially as MathspeedST (2013), followed by a long TPISC series of papers and books (beginning 2014 to present day). TPISC stands for The Pythagorean Inverse Square Connection. What started as a pure math inquiry into the Inverse Square Law lead to the discovery of the BIM (BBS-ISL Matrix). Another right-angle matrix grid -- but not a multiplication table. Although the natural whole number 0,1,2,3,4,5,...populate each axis and the common (prime) diagonal between the two is indeed the square of the axial number, the rest of the cells in the Inner Grid are formed from the difference between the horizontal and vertical intercepts of that prime diagonal. Not only did it reveal all the Primitive Pythagorean Triples on the Active Rows, so too were found all the Primes -- always on the same Active Rows, though independently. The BIM was an extraordinarily rich right-angle matrix grid that continues to shed light in the pure Number Theory discipline.

In 2023, while working on the BIM in writing up what was to be TPISC V, yet another, even more basic right-angle matrix grid was discovered: the Divisor Matrix Table (DMT). And now we are back to a type of multiplication table, only here the top horizontal axis consists of one fractal -- the Butterfly Fractal 1 (BF1) that is simply the result of starting with quantity 1, doubling that to 2, doubling that to 4, to 8, to 16,... -- and the side vertical axis of another fractal -- the ODDs: 1-3-5-7-9... Here we have one fractal at a right-angle relationship to another, giving us the DMT. The DMT is the mother-parent matrix of all the subset child matrix tables, like the BIM, other multiplication tables, and such. It contains ALL the natural whole integer numbers, placed exclusively in one -- and only one -- specific cell location. It also divides those natural numbers into ODDs, EVENS NOT÷4, and EVENS÷4, from left to right columns, respectively. The 1-2-4-8-16... BF1 at the top of the DMT is simply repeated x3-x5-x7-x9-x11... in the Rows as one descends, while at the same time, in going from the ODDs in Col 1, each subsequent Col is simply double the one before. Note that one fractal has a difference (∆) of x2 the previous value, while the other fractal that ∆ is +2 the previous value!

Now in 2026, something truly amazing has been found: Table205! If one takes the DMT and adds +1 to each cell, you get Table205 -- another right-angle matrix table -- that changes everything. Well, almost!

Adding +1 changes every ODD to an EVEN, and, every EVEN to an ODD. So now, former Col 1 (1-3-5-7-9...) is now Col 1: 2-4-6-8-10..., but the subsequent Columns are no longer simply 2x the previous -- now they are (2x previous)-1. Likewise, the former BF1 of Row 1 (1-2-4-8-16...) is now 2-3-5-9-17.... In BOTH tables, the Column values increase by +2-+4-+8-+16-+32, i.e. they both have a ∆ that equals the BF1. But the Row values in the T205 increase by (2x the previous)-1. Thus, in T205, not only are the ODDs now spread out over the Columns -- revealing the Primes, including those extremely rare Mersenne Primes, "on the road to Riemann" -- the ∆'s between Column values follow the fractal BF1 across the table! The BF1 is still there, only now in stealth mode as simply the ∆.

Right-angle relationships are dominant in informing the connections and interconnections of the fundamental notions of quantity: the who, what, where, when, why and how a quantity of ST is generated!

If the first pulse-propagated ST was the Universe, and the 2nd two, the third 4, the fourth 8...ST Universes, one thing is for sure: at every moment the total amount of ST Universes will always be the same -- always conserved -- as the CoST. And to that, we owe the fundamental right-angle relationship between two fundamental fractals -- the EVENs and the ODDs!

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BIM-DMT-T205

Oceans of Numbers: BIM-->DMT-->T205 -- The Road to Riemann

Number have prejudice -- even when they all count.

ODDs vs EVENs

Start with all the ODDs and double them --> you get EVENs. These EVENs are NOt÷4, but if you double them, now the new EVENs are all ÷4

What about them ODDs?

Even they are prejudiced! Some few are Prime, the rest are all composite -- made of other ODDs. (Of course, by definition, all the EVENs are composites, too. And, can all be made from 2 or more Primes.)

Now what?

So place all the ODDs in Column 1, double them to form the next Column, double that Col. to form the next, and so on. What do you get? --> the Divisor Matrix Table (DMT). It contains ALL the natural whole numbers, each entered exactly one time --- and, all of the Divisors, too! It is the great set within the Oceans of Numbers! It puts chaos in the waste bin as ALL numbers are connected!

And where did that come from?

Looking back, we see that the DMT came from an earlier matrix subset, the BIM (BBS-ISL Matrix). The BIM presents most everything that relates to the Inverse Square Law (ISL) -- like the Pythagorean Triples. It is a matrix table of squares -- i.e. Areas -- and as such presents us with a lot of visual geometry! It also deals with Primes, including the Mersenne Primes.

And it is in chasing down those extremely rare Mersenne Primes that leads us on to The Road to *Riemann.

An old quote from college days: "Not only is the future not what it used to be, it's not even WHEN it used to be!

Here is the amazing thing: If one simply adds +1 to each of the values of the DMT, one gets a new, transformative matrix table -- Table205 (T205). In T205 everything is flipped on its head: all the ODDs are now EVENs, and all the EVENs are now ODDs! In doing so, we find that every one of the 52 known Mersenne Primes emanates from the one-and-only EVENs Column that forms Col. 1. As does every Prime!

You've heard of "Roll Over Beethoven," well, now it's time for "Roll Over Riemann" -- the future has arrived!

*Riemann Hypothesis, states that all Primes emanate from the non-trivial, critical line of zeros -- thus giving predictable order to their distribution. See the link for more.