#circular spiral

[rbrooksdesign]

"Circular - BF1,-11", Feb 11, 2026, digital, Reginald Brooks

~spiral around (down) a 2D plane = cone (tetrahedron, if edges remain sharp)

~one can generate a smooth cone while still keeping track of the quadrants that a spiral would encounter

~perhaps a cone with the 4 quadrant edges (lines) surface-marked gives our Running Sums (∑) a place to reside, with our BF1 between

The Butterfly Fractal 1 (BF1): 1-2-4-8-16-...

Double 1 = 2, going clockwise, double 2 = 4, double 4 = 8, double 8 = 16, ...

~one quadrant at a time,

[rbrooksdesign]

"Circular - BF1, -13", Feb 13, 2026, digital, Reginald Brooks

~spiral around (down) a 2D plane = cone (tetrahedron, if edges remain sharp)

~one can generate a smooth cone while still keeping track of the quadrants that a spiral would encounter

~perhaps a cone with the 4 quadrant edges (lines) surface-marked gives our Running Sums (∑) a place to reside, with our BF1 between

The Butterfly Fractal 1 (BF1): 1-2-4-8-16-...

The Running Sums (∑) -- 1+2=3 3+4=7 7+8=15...

[rbrooksdesign]

"Circular - BF1, -19", Feb 19, 2026, digital, Reginald Brooks

Serendipty

So often rings a fine truth:

A good cup of tea

~~~

28 is a Perfect Number -- 1+2+4+7+14=28 -- and lies on ODD Row 7. 7 is a Prime, but also a very rare Mersenne Prime (Mp). A Mp is a prime that is 1 less than an Exponential Power of 2 -- 2ᵖ-1.

ALL the EVENs from the 3rd Column over -- ALL being ÷4 -- when sequentially divided by 2 -- eventually fall into the 2nd Column over, where NO EVEN is ÷4.

Dividing the numbers in Column 2 by 2, ALWAYS ends with an ODD in Column 1.

That, in essence, is the Divisor Matrix Table (DMT).

"Cones" represent 1 Row on the DMT.

EVENs ÷4 are L & R, here.

EVENs ÷4 are F&B.

ODDs, on top.

EVENs twist.

Down.

C.

Enjoy.

Ride rewards.

A good cup of tea.

[rbrooksdesign]

"Circular - BF1,-22 ", Feb 22, 2026, digital, Reginald Brooks

"Cone" and DMT Row 31 (ORANGE) contain the Perfect Number (PN) 496. 31 is both a Prime, and a subset Mersenne Prime (Mp).

"Cone" and DMT Row 33 (GREEN-BG) contains NO PN's. 33 is not Prime.

"Cone" and DMT Row 35 (PURPLE-BG) contains NO PN's. 35 is not Prime.

...and so it goes. The vast majority of "cones" and DMT Rows are not Prime and less so Mp. The next TRUE Mp "cone" and DMT Rows are: 127--8191! The gap widens.

...and every one of the DMT Rows is but a multiple of the Butterfly Fractal 1 (BF1) Row 1.

[rbrooksdesign]

"Circular - BF1, -16", Feb 16, 2026, digital, Reginald Brooks

~spiral around (down) a 2D plane = cone (tetrahedron, if edges remain sharp)

~one can generate a smooth cone while still keeping track of the quadrants that a spiral would encounter

~perhaps a cone with the 4 quadrant edges (lines) surface-marked gives our Running Sums (∑) a place to reside, with our BF1 between

???

Each of the "cones" above is but one horizontal line below. What is it forming? The Divisor Matrix Table (DMT), the universal table that contains ALL natural whole numbers and their divisors.

"Duck Soup!" Well, maybe...

Converting the Circular - BF1 spiral imagery into its component parts -- multiplies of the fundamental BF1 fractal -- gives one a new perspective. Each "cone" -- as a BF1 multiple -- starts out as an ODD, then doubles and doubles and doubles as it traces out a clockwise spiral path down and around the "cone." Each "cone" is a Row on the Divisor Matrix Table (DMT). ALL the numbers after the initial starting ODD are EVENs.

The first EVEN on the "cones" and the first EVEN Column on the DMT contain EVENs NOT ÷ 4. All other EVENS thereafter are ÷4 and appear on the "cones," and, on all the next Columns on the DMT.

On the "cones," we see that all the numbers going down one side/face end with the same "ones" number. On the DMT this occurs at every 4th number across the Rows. This becomes important when -- aside from the *one exception at the start -- ALL Perfect Numbers will fall on either the LEFT or RIGHT face of the "cones" and only in Set 1 Columns of the DMT.

*one exception: Perfect Number 6 (where its factors 1-2-3 sum up to equal 6), is the ONLY PN found on the EVENs NOT ÷4 Column (Set 2) on the DMT -- and seen on the BLUE "cone."

You can handle the fractal "cones" in your hands, so to speak, but the DMT locks their entanglement into place.

[rbrooksdesign]

"Circular - BF1, -10", Feb 10, 2026, digital, Reginald Brooks

~spiral around (down) a 2D plane = cone (tetrahedron, if edges remain sharp)

~one can generate a smooth cone while still keeping track of the quadrants that a spiral would encounter

~perhaps a cone with the 4 quadrant edges (lines) surface-marked gives our Running Sums (∑) a place to reside, with our BF1 between

[rbrooksdesign]

"Circular - BF1, -18", Feb 18, 2026, digital, Reginald Brooks

~spiral around (down) a 2D plane = cone (tetrahedron, if edges remain sharp)

~one can generate a smooth cone while still keeping track of the quadrants that a spiral would encounter

~perhaps a cone with the 4 quadrant edges (lines) surface-marked gives our Running Sums (∑) a place to reside, with our BF1 between

???

Each of the "cones" above is but one horizontal line below. What is it forming? The Divisor Matrix Table (DMT), the universal table that contains ALL natural whole numbers and their divisors.

~~~

"Duck Soup!" Well, maybe...

We now move on to the next 3 "cones" that are, indeed, the next 3 Rows on the DMT. The second Perfect Number (PN) is revealed: PN 28.

[rbrooksdesign]

"Circular - BF1, -15", Feb 15, 2026, digital, Reginald Brooks

~spiral around (down) a 2D plane = cone (tetrahedron, if edges remain sharp)

~one can generate a smooth cone while still keeping track of the quadrants that a spiral would encounter

~perhaps a cone with the 4 quadrant edges (lines) surface-marked gives our Running Sums (∑) a place to reside, with our BF1 between

???

Each of the "cones" above is but one horizontal line below. What is it forming? The Divisor Matrix Table (DMT), the universal table that contains ALL natural whole numbers and their divisors.

What is the DMT based on? The first cone. It shows the Butterfly Fractal 1 (BF1) as it spirals clockwise around -- and down -- the cone, advancing one number per quadrant, starting with "1" first at the center and with "2" in the facing quadrant.

What is the BF1? It is a primordial -- if you will -- fractal that starts with quantity "1" and doubles each successive number: 1-2-4-8-16-32-64-... Mathematically, it is represented by the successive Exponential Power of 2 values: 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, ...

Why is it a fractal? It is a self-similar, re-iterative expression of the whole in every part.

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Why do the next "cones" -- and DMT Rows -- start with the ODD numbers? The BF1 EVENs simply repeat themselves, the ODDs give new multiples of the BF1 + any and all of the other EVENS not found in the BF1 . Together, the BF1 x ODDs gives ALL the natural whole numbers.

So what is the point of showing that the DMT can also be visualized as a Circular-BF1 spiral -- first -- and sub-sequentially as a series of "cones" -- second? ...next time

And, what about all the hullabaloo about the Running Sums (∑)? ...next time (hint: starting with "1", go up the fractal diagram, adding the sum of each Row: 1-3-7-15-31-)