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I couldn't find a place to post this thread, so it will have to go here for the moment. It followed from a discussion about pixellated Sierpin ski triangles. I did some work on them this morning and came up with what I believe to be the closest approximation in numerical geometry to a Sierpinski triangle in plane geometry.

I started with triangle 1 (T1) and kept on doubling to give T2, T4, T8, T16, T32, T64 . . .

It has several interesting features

1. It corresponds to removing all the even numbers in Pascal's triangle.

2. The first triangle from which a unit could be removed was T4, giving a crude Sierpinski triangle of 9 units. The triangles created thereafter have multiples of 3 units: 27, 81, 243, 729 . . . .

3. The index of the largest triangle that can be cut from the centre at each stage corresponds to the number counts down to each row of Pascal's triangle (details in the image).

4. The perfect numbers are all accommodated and I show the first three: 6, 28, 496. The next one is 8128, which the next stage produces.

5. The index of each triangle corresponds to the row counts in Pascal's triangle (details in the image).

6. These are also the Mersene numbers (2^n - 1): 1, 3, 7, 15, 31.

7. The first three perfect numbers correspond to three Mersene primes 3, 7, 31. The next one corresponds to another Mersene prime, 127 (T127 is 8128).
Note: I just checked and the next perfect number beyond 8128 is 33550336, which like all of them is triangular. Its base is 8191, also a Mersene orime. So I’m guessing that all perfect numbers will fit onto this template, with enough doublings, and have a prime number index.p (I’m also guessing this is well known among number theorists!) I must check and see if that has been proven.
Note 2: I was right. It’s a well known result in number theory. Nice to see it visually though, and from the simplest of beginnings.

8. I believe the only complete Sierpinski triangle will be cut from T(infinity), but that's just an amateur's intuitive guess.

9. The area of the Sierpinski triangle after infinite iterations is zero, and so the fraction no. units ST/no. units T will tend to zero as the sequence progresses. The first few ratios are 0.9, 0.75, 0.595.., 0.460.., 0.350.., 0.264..,

Anyway, apologies for any lack of mathematical formalism. I just thought it was interesting and some of you might like to see the first few natural Sierpinski triangle approximations. There are other types, but this is the closest to the fractal in plane geometry,

It was worth doing for itself, but also because I believe I have found two Sierpinski triangles (though not in this 'ideal' sequence) within the Hebrew Bible's first verse, as a mirrored pair.

bluetriangle wrote: Tue Mar 03, 2026 7:38 am I couldn't find a place to post this thread, so it will have to go here for the moment. It followed from a discussion about pixellated Sierpinski triangles. I did some work on them this morning and came up with what I believe to be the closest approximation in numerical geometry to a Sierpinski triangle in plane geometry.

I started with triangle 1 (T1) and kept on doubling to give T2, T4, T8, T16, T32, T64 . . .

It has several interesting features

1. It corresponds to removing all the even numbers in Pascal's triangle.

2. The first triangle from which a unit could be removed was T4, giving a crude Sierpinski triangle of 9 units. The triangles created thereafter have multiples of 3 units: 27, 81, 243, 729 . . . .

3. The index of the largest triangle that can be cut from the centre at each stage corresponds to the number counts down to each row of Pascal's triangle (details in the image).

Fabulous post Bill! This is a fine place for it, though I guess I could start a new Pure Math subforum (might be a good idea, but it would probably always digress to gematria given the interests of folks on this forum).

You really laid out the most significant facts relating to the sequence of finite Sierpinski triangles. I particularly like the fact you noted the connection to the modular (base 2 = even/odd) Pascal triangle. Similar fractal-like patterns appear no matter what modulo we use.

I am particularly pleased with the way it visually represents the exponential growth of powers of 3 as triangles within triangles. That feels so natural and beautiful. Powers of 3 play a big role in the alphanumeric structure of Genesis as should be clear given the appearance of hte 3rd cube in the GenSet and it's geometric relation to the 4th hexagon H(4) = 37 = 64 - 27 = 4^3 - 3^3.

I discuss cubic projections and the GenSet here: https://www.biblewheel.com/GR/GR_HexProjections.php

Thanks. I did it after our discussion of these figures, and also because I find it fascinating, as Mr. Spock would say. I added a couple,e of notes you probably missed. Essentially the relationship between the Mersene primes and the perfect number indices is well known in number theory.

I was playing around with it and noticed that the gaps between perfect numbers in this figure correspond to the numbers down the spine of Pascal’s triangle: 1, 2, 6, . . . The next number down the spine is 20 and I wonder if this is the gap in triangular segments between 33550336 and the next perfect number.
Note: It isn't. But then there is no definite pattern to the primes and I believe that would have been one.

The number of triangular gaps at each iteration, 1, 4, 13, 40, 121, 364. . . is sequence https://oeis.org/A003462 which is a(n) = (3^n -1)/2. It was first noted in 2004.

bluetriangle wrote: Tue Mar 03, 2026 8:52 am Thanks. I did it after our discussion of these figures, and also because I find it fascinating, as Mr. Spock would say. I added a couple,e of notes you probably missed. Essentially the relationship between the Mersene primes and the perfect number indices is well known in number theory.

I was playing around with it and noticed that the gaps between perfect numbers in this figure correspond to the numbers down the spine of Pascal’s triangle: 1, 2, 6, . . . The next number down the spine is 20 and I wonder if this is the gap in triangular segments between 33550336 and the next perfect number.

No, I didn't miss your references to Mersenne primes and perfect numbers. Just didn't have time to comment on everything I saw.

Again, let me thank you for your excellent post. It contributes greatly to this glorious study of the infinite Wisdom God encoded in His Holy Word.

And thank you for being patient with me as I test everything according the light He has given me.

As for the next perfect number, they've searched up to some huge number, probably bigger than what you've suggested.

God bless you my friend!