#euclid euler

[rbrooksdesign]

"Circular - BF1, 15", Jan 15, 2026, digital/acrylic, Reginald Brooks

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

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

With exception of x=2, all other Perfect Number x's are in the vertical quadrants, as 2ᵖ⁻¹ .

With exception of z=3, all other Perfect Number z's are in the horizontal quadrants as 2ᵖ -1.

The Perfect Number (PN) = xz = 2ᵖ⁻¹ (2ᵖ -1). PN=2ᵖ⁻¹ (2ᵖ -1) is the Euclid-Euler Theorem.

p=prime, and when that prime is a Mersenne Prime, xz=PN.

A Mp is a prime that is 1-less than an exponential power of 2; e.i. p=Mp=2³ -1= 8-1=7.

The Running Sums (∑) -- 1+2=3 3+4=7... = the EVENs in the horizontal quadrants - 1. (WHITE numbers on the diagonals)

Examples:

ADD 3 + 4 = 7, where the 3 = ∑ = z (previous) and 4 = x. Multiply 4•7 = 28 = xz = PN.

ADD 15 + 16 = 31, where the 15 = ∑ = z (previous) and 16 = x. Multiply 16•31 = 496 = xz = PN.

ADD 63 + 64 = 127, where the 63 = ∑ = z (previous) and 64 = x. Multiply 64•127 = 8128 = xz = PN.

The TRUE Mp-PNs have the BLACK "x's" outlined in white -- all others are "containers" that follow the pattern.

[rbrooksdesign]

"Circular - BF1, 5", Jan 5, 2026, digital/acrylic, Reginald Brooks

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

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

With exception of x=2, all other Perfect Number x's are in the vertical quadrants, as 2ᵖ⁻¹ .

With exception of z=3, all other Perfect Number z's are in the horizontal quadrants as 2ᵖ -1.

The Perfect Number (PN) = xz = 2ᵖ⁻¹ (2ᵖ -1). PN=2ᵖ⁻¹ (2ᵖ -1) is the Euclid-Euler Theorem.

p=prime, and when that prime is a Mersenne Prime, xz=PN.

A Mp is a prime that is 1-less than an exponential power of 2; e.i. p=Mp=2³ -1= 8-1=7.

Think about the Running Sums (∑) -- 1+2=3 3+4=7... they are kind of hidden until you see them. Coming up.

May you see, always!

[rbrooksdesign]

"Circular - BF1, 14", Jan 14, 2026, digital/acrylic, Reginald Brooks

The Butterfly Fractal 1 (BF1): 1-2-4-8-16-... (BLACK numbers = x, some outlined in white)

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

With exception of x=2, all other Perfect Number x's are in the vertical quadrants, as 2ᵖ⁻¹ .

With exception of z=3, all other Perfect Number z's are in the horizontal quadrants as 2ᵖ -1.

The Perfect Number (PN) = xz = 2ᵖ⁻¹ (2ᵖ -1). PN=2ᵖ⁻¹ (2ᵖ -1) is the Euclid-Euler Theorem.

p=prime, and when that prime is a Mersenne Prime, xz=PN.

A Mp is a prime that is 1-less than an exponential power of 2; e.i. p=Mp=2³ -1= 8-1=7.

The Running Sums (∑) -- 1+2=3 3+4=7... = the EVENs in the horizontal quadrants - 1. (WHITE numbers on the diagonals)

Examples:

ADD 3 + 4 = 7, where the 3 = ∑ = z (previous) and 4 = x. Multiply 4•7 = 28 = xz = PN.

ADD 15 + 16 = 31, where the 15 = ∑ = z (previous) and 16 = x. Multiply 16•31 = 496 = xz = PN.

ADD 63 + 64 = 127, where the 63 = ∑ = z (previous) and 64 = x. Multiply 64•127 = 8128 = xz = PN.

The TRUE Mp-PNs have the BLACK "x's" outlined in white -- all others are "containers" that follow the pattern.

[rbrooksdesign]

"Circular - BF1, 2", Jan 2, 2026, digital/acrylic, Reginald Brooks

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

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

With exception of x=2, all other Perfect Number x's are in the vertical quadrants, as 2ᵖ⁻¹ .

With exception of z=3, all other Perfect Number z's are in the horizontal quadrants as 2ᵖ -1.

The Perfect Number (PN) = xz = 2ᵖ⁻¹ (2ᵖ -1).

p=prime, and when that prime is a Mersenne Prime, xz=PN, e.i. PN=4*7=28 where p=3.

A Mp is a prime that is 1-less than an exponential power of 2; e.i. p=Mp=2³ -1= 8-1=7.

[rbrooksdesign]

"Circular - BF1, 6", Jan 6, 2026, acrylic, Reginald Brooks

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

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

With exception of x=2, all other Perfect Number x's are in the vertical quadrants, as 2ᵖ⁻¹.

x is the short side of the PN rectangle. z is the long side (2ᵖ -1). PN=xz.

With exception of z=3, all other PN z's are in the horizontal quadrants as 2ᵖ -1.

As we will also soon see, the Running Sums (∑) of the BF1 will also generate the same z's.

While the Euclid-Euler Theorem gives PN=2ᵖ⁻¹ (2ᵖ -1), where p=prime, it is easier to see the patterns when x and z are substituted and the exponent is change to n, as n=0, 1, 2, 3,...

Note: as seen in this painting, sometimes one serendipitously bumps the strict sequential placement of the BF1. Perhaps indicative of a spiral sequence count!

---------x--------

| |

| |

z PN=xz z

| |

| |

---------x--------

[rbrooksdesign]

"Circular - BF1, 4", Jan 4, 2026, digital/acrylic, Reginald Brooks

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

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

With exception of x=2, all other Perfect Number x's are in the vertical quadrants, as 2ᵖ⁻¹ .

With exception of z=3, all other Perfect Number z's are in the horizontal quadrants as 2ᵖ -1.

The Perfect Number (PN) = xz = 2ᵖ⁻¹ (2ᵖ -1). PN= 2ᵖ⁻¹ (2ᵖ -1) is the Euclid-Euler Theorem.

p=prime, and when that prime is a Mersenne Prime, xz=PN.

A Mp is a prime that is 1-less than an exponential power of 2; e.i. p=Mp=2³ -1= 8-1=7.

Think about the Running Sums (∑) -- 1+2=3 3+4=7... they are kind of hidden until you see them. Coming up.

[rbrooksdesign]

"Circular - BF1, 1," Jan 1, 2026, acrylic, Reginald Brooks

(Yes, finished painting it on 12-31-25, tweaked it later on---it's going in today!)

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

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

With exception of x=2, all other Perfect Number x's are in the vertical quadrants, as 2ᵖ⁻¹ .

With exception of z=3, all other Perfect Number z's are in the horizontal quadrants as 2ᵖ -1.

The Perfect Number (PN) = xz = *2ᵖ⁻¹ (2ᵖ -1).

p=prime, and when that prime is a Mersenne Prime, xz=PN.

A Mp is a prime that is 1-less than an exponential power of 2; e.i. p=Mp=2³ -1= 8-1=7.

*2ᵖ⁻¹ (2ᵖ -1)=PN is the Euclid-Euler Theorem.

Sent from my iPad

[rbrooksdesign]

"DMT-EDMT Series for December Geometry #5"

The Butterfly Fractal 1 (BF1) is a fractal resulting from the simple doubling of quantity "1" -- 1 -- 1x2=2 -- 1x2x2=4. -- 1x2x2x2=8,...

When you lay it out into its "Butterfly-like" array, you can see that every "1" becomes a new source for the self-similar, re-iterative redundancy of the same "fractal" pattern within.

The BF1 pattern is repeated on both sides (of the "wings.")

The new source is circled in RED.

Counting the number of REDs at the bottom = 2, that is also the value of "x" ("x" is the short side of the PN rectangle, xz. z=Mp=the long side.)

Another tidbit that plays out: p=2, is the number of levels up, starting with level one encircled RED.

Math: Euclid-Euler Theorem = 2ᵖ⁻¹ (2ᵖ -1) = Perfect Number (PN) where 2ᵖ⁻¹ = x = 2¹ =2, and, 2ᵖ -1 = z = 3 = Mersenne Prime (Mp), as xz=PN.