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RAMcGough wrote: Sat Feb 28, 2026 12:33 pm Thank you for that detailed explanation. Seeing the numbers that you actually used really helps. It was hard on my old eyes to try to count those dots. It would be good if you added a note to the images so people like me can understand what you really did and verify the numbers.

It's been hard on my old eyes too! In fact it's been hard on every part of me. I'm like an old horse ready for the knacker's yard now (I'm only 66, but feel older). But I'll bear your words in mind. Thanks for letting me know.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pmI agree that your method of recursively removing the "largest possible" inverted triangle is logically consistent and it does create an approximation of a Sierpinski triangle. But the visual and mathematical inconsistency remains; some dots are shared while others are not. This is just a fact because an arbitrary triangular grid does not accommodate a Sierpinski triangle. Yes, it visually reminds one of a Sierpinski triangle, but close inspection shows it is not really a fractal.

It will only ever be an approximation of a fractal, yes. But the larger the numerical triangle the closer we will come to that fractal. So T(Infinity) will be the only numerical triangle that can also become a perfect fractal, after infinite iterations!

The logos star is the same. It's a second-iteration Koch snowflake, but in reality is just a crude approximation of the true fractal. Again only T(Infinity) will give the actual fractal. Only G-triangles can be taken one or more iterations towards the Koch snowflake and most of them only one iteration at that. The first that can be taken two iterations is T91. The first that can be taken three iterations is, oh I forget, but it yields a snowflake with 1261 counters.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm I think it may be the best approximation you could get if you start with an arbitrary triangle T(n). But to me, such approximations are not worthy of God.

I'm not trying to give offense, but perhaps it's what you expect from God that's the issue.

Taking the encoding of pi and e in G1.1 and J1.1, these are transcendentals and so impossible to encode using numbers or algebraic formulae. Even if you use a Taylor series, such as

e = 0! + 1! + 2! + 3! . . . . . .

you would need an infinitely long book to do so. They are good approximations of the numbers though and of course encoding them was far beyond the capabilities and knowledge of the scribes. If they weren't encoded by The Logos how could it have been done? Advanced aliens with a quantum computer? Given the huge timescales and the different physical locations, languages and cultures involved, as well as the myriad other codes in the Bible, that can be discounted. It can only have come from a realm beyond time and place, something like Bohm's implicate order (the Mind of God), manifesting over time in the explicate order (the material universe).

Perhaps the accuracy is also a reflection of our own sin. Maybe if man was a better creature the codes would be better. By this I mean that the codes are like standing waves created by our interaction with the word. A good analogy is standing waves in water or similar media. If our minds were purer God could have done more with them - and vice versa of course.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pmIt reminds me of the many people who take approximations of real or rational numbers and round them off to fit integer patterns they like.

Such as this?
1080/1480 gives alpha x 100, with an error only 1 in 132,000. It's meaningful too, given that alpha qualifies the strength of the interaction between light and matter. 1080 is the Holy Spirit and 1480 is the Incarnation: light and matter. Then there is 2368/754 approximating to pi and 2368/749 approximation to the square root of 10. I think they're genuine and the pi one is also meaningful, as Jesus is the mediator between heaven (circle) and earth (straight line, part of a square). .

RAMcGough wrote: Sat Feb 28, 2026 12:33 pmOr when people use that algorithm that produces the first few digits of pi from Genesis 1:1 and e from John 1:1. Maybe God did it, but it has never felt convincing to me. I grant that it is intriguing that a single algorithm produces those two numbers from those two verses which are related in other more obvious ways, so I don't reject it out of hand. It's just not my cup of tea.

Well, I've just written about that. It's my cup of tea, certainly. Incidentally, when the alpha encoding, (27013627)^2, is added to the pi and e encodings, the final error is only 1 part in 16 million or thereabouts.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pmHere's the big question: Where's the motivation for thinking that "only begotten Son" and "The Holy Spirit" should be encoded this way? It has the classic hallmarks of the game of creating a new pattern from an established pattern and then scanning the entire universe of possible words and phrases that might fit something you like. The phrases you chose have absolutely nothing to do with the verse itself or the Sierpinski triangle. This is why it looks like a combination of cherry picking and force fitting a pattern.

I just answered that in my previous post. I had to cherry pick from the start. Both numbers leapt out at me and are part of a pattern of Divine signatures found in many preeminent locations. I believe I've found many of them, including little ELS codes in the NIV. Here's one. This works better with Courier.

IN
THEBEGIN
NINGGODC
REATEDTH
EHEAVENS

Do you see God crossing God here? The skip interval is 8 too.

I think our approaches differ in that you are more deductive and I'm more inductive. I worked in labs for 29 years (developing cleaning and hygiene products) and used that kind of reasoning all the time. In many ways I've had to live with uncertainty and perhaps it helped me.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm I respect your efforts. Please don't take this as my "final judgment". I'm merely telling you what I see as I see it. I was ignorant of the exact details of your method when I first commented, and your explanation cleared things up a lot, so I hope you see that's how all these conversations will go. You are obviously very intelligent and careful in your presentations and I really enjoy talking with you. So once again, please have patience with me as we work together to come to a common understanding.

God bless you my friend,

God bless you too. And God willing we'll climb the mountain of knowledge and meet at the summit, where all is known.

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm It's been hard on my old eyes too! In fact it's been hard on every part of me. I'm like an old horse ready for the knacker's yard now (I'm only 66, but feel older). But I'll bear your words in mind. Thanks for letting me know.

I'm 66 too. My body is slowing down, but my spirit feels fresh like a child since coming back to faith in Christ. I feel like I'm splashing around in a fountain of love, joy and peace - the fruit of the Spirit.

I've been riding my bike almost every day for many years, and I'm watching my average speed drop from 17 mph, to 15 mph, to 14 mph. And I've had to start wearing reading glasses.

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm It will only ever be an approximation of a fractal, yes. But the larger the numerical triangle the closer we will come to that fractal. So T(Infinity) will be the only numerical triangle that is also a perfect fractal!

The logos star is the same. It's a second-iteration Koch snowflake, but in reality is just a crude approximation of the true fractal. Again only T(Infinity) will give the actual fractal. Only G-triangles can be taken one or more iterations towards the Koch snowflake and most of them only one iteration at that. The first that can be taken two iterations is T91. The first that can be taken three iterations is, oh I forget, but it yields a snowflake with 1261 counters.

You are correct that a true fractal is only in limit as n => infinity. But the term "fractal" is also used (loosely) for the sequence of iterations that form the fractal in the limit. The Logos Star is an EXACT iteration in the Koch sequence. The Sierpinski triangles that you create from T(73) are NOT an exact iteration in that sequence. The Sierpinski sequence is defined by triangles T(n) where n = 2^n + 1 and yields the sequence

3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575, 265722 ... https://oeis.org/A067771

The triangle T(73) plays no role in that sequence.

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm I think it may be the best approximation you could get if you start with an arbitrary triangle T(n). But to me, such approximations are not worthy of God.

I'm not trying to give offense, but perhaps it's what you expect from God that's the issue.

Taking the encoding of pi and e in G1.1 and J1.1, these are transcendentals and so impossible to encode using numbers or algebraic formulae. Even if you use a Taylor series, such as

e = 0! + 1! + 2! + 3! . . . . . .

you would need an infinitely long book to do so. They are good approximations of the numbers though and of course encoding them was far beyond the capabilities and knowledge of the scribes. If they weren't encoded by The Logos how could it have been done? Advanced aliens with a quantum computer? Given the huge timescales and the different physical locations, languages and cultures involved, as well as the myriad other codes in the Bible, that can be discounted. It can only have come from a realm beyond time and place, something like Bohm's implicate order (the Mind of God), manifesting over time in the explicate order (the material universe).

Perhaps the accuracy is also a reflection of our own sin. Maybe if man was a better creature the codes would be better. By this I mean that the codes are like standing waves created by our interaction with the word. A good analogy is standing waves in water or similar media. If our minds were purer God could have done more with them - and vice versa of course.

Please don't worry about giving offense. I'll let you know if my feathers get ruffled. I'm loving this conversation. And I have nothing to defend. I'm just trying to discover the truth with you. I appreciate your help.

I'm glad we're reviewing this. I haven't given this particular pattern any thought since my restoration last February. I'm seeing everything with new eyes, so let's review this together and see what we see.

The algorithm gives only the first 4 digits for pi and for e. At first glance, that doesn't seem like much. If God did this, it obviously has nothing to do with limitations imposed by the transcendental nature of the numbers. He didn't even try to give a significant approximation like 10 or 20 digits, so their transcendence seems totally irrelevant to me. But thinking of the probabilities and the mathematical centrality of the two numbers is making me think this may be significant. There are 9900 four digit numbers that don't start with 0. Thus, the random chance of getting the first four digits of pi is 1/9900 and the same for e. Now if these were just two random numbers with no intrinsic relation or mathematical preeminence, then it might not be meaningful to take the JOINT PROBABILTY of finding both, since the researcher may just as well have found some other random number. But these numbers are premier mathematical constants that are UNITED in one of the most famous and beautiful mathematical relations in existence, the Euler Identity: And this unique pair (pi & e) are derived from the same algorithm applied to the two premier creation passages of Scripture which are themselves UNITED through an amazing and abundant set of algebraic/geometric relations:

Genesis 1:1 = 2701 = 37 x 73 (mirrored primes)
John 1:1 = 3627 = 39 x 93 (mirrored semiprimes)

Geneses 1:1 + John 1:1 = T(112) = 6328

Etc. etc., etc.

Therefore, the JOINT PROBABILITY is significant, and it is (1/9900)^2 = 1 in 98 million.

I now feel this is very significant. I don't know why I didn't see it before. Thanks for reviewing it with me!

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm

RAMcGough wrote: Sat Feb 28, 2026 12:33 pmIt reminds me of the many people who take approximations of real or rational numbers and round them off to fit integer patterns they like.

Such as this?
1080/1480 gives alpha x 100, with an error only 1 in 132,000. It's meaningful too, given that alpha qualifies the strength of the interaction between light and matter. 1080 is the Holy Spirit and 1480 is the Incarnation: light and matter. Then there is 2368/754 approximating to pi and 2368/749 approximation to the square root of 10. I think they're genuine and the pi one is also meaningful, as Jesus is the mediator between heaven (circle) and earth (straight line, part of a square). .

I'm glad you understand the meaning of alpha as the strength of interaction between light and matter. That's why it's appearance as a large prime factor in John 1:1-5 stuck me as so significant. It is reiterated in the passage (see here). We have:

All things were made by him = 2055 = 15 x 137
Sum of John 1:1-5 = 26715 = 13 x 15 x 137

Look at the right side, where the number 137 connects to the two parts:


Now to be clear, I am not adverse to the idea of comparing ratios in general. I've just seen it used a lot in ways that seemed to be unjustified cherry picking. I myself derived the GenSet from the continued fraction of inverse alpha. But note - every line is EXACT. The convergents are approximations, but the values of the convergents are calculated EXACTLY in terms of the GenSet numbers.

Fine Structure Constant
Now as for your ratio of 1080/1480. That's "nice" but two problems. You are using To Pneuma Hagion = 1080 for the numerator but just Christos = 1480 for the denominator. That gives an uncomfortable tension of a ratio between an arthrous title to an anarthrous title. Also, the reduced fraction is 1080/1480 = 27/37 which are two genset numbers. So the real significance comes from the genset which is derived from a coherent analysis of Genesis and John and integrates with the continued fraction of inverse alpha, not by comparing arthrous and anarthrous titles with little motivation from Scripture. I grant that comparing Spirit to Christ (incarnation) has a nice ring to it, but it seems arbitrary and subjective, which is what I avoid in all my research.

Please forgive me brother if my words sound sharp! I'm just speaking from the heart with my mind on fire!

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm
God bless you too. And God willing we'll climb the mountain of knowledge and meet at the summit, where all is known.

Amen! It's a joy to be climbing the Mountain of God with you.

RAMcGough wrote: Sat Feb 28, 2026 7:31 pm

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm It's been hard on my old eyes too! In fact it's been hard on every part of me. I'm like an old horse ready for the knacker's yard now (I'm only 66, but feel older). But I'll bear your words in mind. Thanks for letting me know.

I'm 66 too. My body is slowing down, but my spirit feels fresh like a child since coming back to faith in Christ. I feel like I'm splashing around in a fountain of love, joy and peace - the fruit of the Spirit.

I've been riding my bike almost every day for many years, and I'm watching my average speed drop from 17 mph, to 15 mph, to 14 mph. And I've had to start wearing reading glasses.

I used to jog regularly, but I gave that up recently. I still walk a lot though. My eldest daughter and I will be tackling Ben Nevis next week. It's Scotland and the UK's highest mountain at 4409ft.

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm It will only ever be an approximation of a fractal, yes. But the larger the numerical triangle the closer we will come to that fractal. So T(Infinity) will be the only numerical triangle that can also become a perfect fractal (after infinite iterations)l

I didn't write this correctly, so I've corrected it.

RAMcGough wrote: Sat Feb 28, 2026 7:31 pm

bluetriangle wrote: Sat Feb 28, 2026 3:52 pm The logos star is the same. It's a second-iteration Koch snowflake, but in reality is just a crude approximation of the true fractal. Again only T(Infinity) will give the actual fractal. Only G-triangles can be taken one or more iterations towards the Koch snowflake and most of them only one iteration at that. The first that can be taken two iterations is T91. The first that can be taken three iterations is, oh I forget, but it yields a snowflake with 1261 counters.

You are correct that a true fractal is only in limit as n => infinity. But the term "fractal" is also used (loosely) for the sequence of iterations that form the fractal in the limit. The Logos Star is an EXACT iteration in the Koch sequence. The Sierpinski triangles that you create from T(73) are NOT an exact iteration in that sequence. The Sierpinski sequence is defined by triangles T(n) where n = 2^n + 1 and yields the sequence

3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575, 265722 ... https://oeis.org/A067771

The triangle T(73) plays no role in that sequence.

Yes, I understand what you mean. We may be at cross purposes here and I wasn't being precise. I need to work out some figures and get back to you on this. The amount of research needed to pin these things down is incredible and as far as I can tell, no one has ever worked these things out before. It's a PhD research project and I'm just an amateur trying my best.

Incidentally, there will be several new numerical sequences to be found here and in the Koch snowflake/antisnowflake sequences. John Elias created sequences from some of the snowflakes I'd already found and published them on the OEIS. But there are many more.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm I'm glad we're reviewing this. I haven't given this particular pattern any thought since my restoration last February. I'm seeing everything with new eyes, so let's review this together and see what we see.

The algorithm gives only the first 4 digits for pi and for e.

It's 5 for pi, but that's a minor point. The errors are roughly the same: 1 in 88000 and 1 in 82000.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm At first glance, that doesn't seem like much. If God did this, it obviously has nothing to do with limitations imposed by the transcendental nature of the numbers. He didn't even try to give a significant approximation like 10 or 20 digits, so their transcendence seems totally irrelevant to me.

Okay, fair enough. I was just making the point (more for others) that it can never be done even in principle.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm But thinking of the probabilities and the mathematical centrality of the two numbers is making me think this may be significant. There are 9900 four digit numbers that don't start with 0. Thus, the random chance of getting the first four digits of pi is 1/9900 and the same for e. Now if these were just two random numbers with no intrinsic relation or mathematical preeminence, then it might not be meaningful to take the JOINT PROBABILTY of finding both, since the researcher may just as well have found some other random number. But these numbers are premier mathematical constants that are UNITED in one of the most famous and beautiful mathematical relations in existence, the Euler Identity:
image.png

My favourite equation! I feel awestruck just contemplating it. And of course it incorporates the fundamental unit 1 and the empty quantity 0, which together give us decimal notation and binary code, and the mysterious i, the basis of complex numbers and a mathematical sleight-of-hand that is nevertheless expressed in the world around us. I think more than anything this convinces me that our material world has mathematical roots.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm And this unique pair (pi & e) are derived from the same algorithm applied to the two premier creation passages of Scripture which are themselves UNITED through an amazing and abundant set of algebraic/geometric relations:

Genesis 1:1 = 2701 = 37 x 73 (mirrored primes)
John 1:1 = 3627 = 39 x 93 (mirrored semiprimes)

Geneses 1:1 + John 1:1 = T(112) = 6328

Etc. etc., etc.

Therefore, the JOINT PROBABILITY is significant, and it is (1/9900)^2 = 1 in 98 million.

Yes and of course they sum to give a number that is in error of the true value of pi + e by only 1 in 850,000. As I said before, when you add alpha, the error then reduces to 1 in around 16 million, another sign they were consciously encoded. It's a beautiful, self-contained masterpiece of design.

Regarding 6328, there is much more to be found by combining these two verses. I've done a suite of pages exploring the geometry that emerges when G1.1 and J1.1 are combined.The figure below is one of them, a third-step Koch snowflake created from G1.1 + the third clause of J1.1. Note that one way it can be sectioned off internally, giving 373 x 7 and 151 x 6, is actually encoded in word strings.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm I'm glad you understand the meaning of alpha as the strength of interaction between light and matter. That's why it's appearance as a large prime factor in John 1:1-5 stuck me as so significant. It is reiterated in the passage (see here). We have:

All things were made by him = 2055 = 15 x 137
Sum of John 1:1-5 = 26715 = 13 x 15 x 137

Look at the right side, where the number 137 connects to the two parts:


Now to be clear, I am not adverse to the idea of comparing ratios in general. I've just seen it used a lot in ways that seemed to be unjustified cherry picking. I myself derived the GenSet from the continued fraction of inverse alpha. But note - every line is EXACT. The convergents are approximations, but the values of the convergents are calculated EXACTLY in terms of the GenSet numbers.

Fine Structure Constant

image.png

This is incredible. I don't know if I've seen this before. What do you get when you take the continued fraction to infinity?

Eddington came up with an equation, of which alpha is a root, that gave an almost exact value:

x^4 - 136x^3 - 136x^2 - 818x + 1 = 0

giving 1/alpha = 137.03599916. . .

This differs from the most recent measurements of alpha only at the 8th decimal digit.

1/alpha = 137.03599917. . .

It might be interesting to compare it with yours for accuracy.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pm Now as for your ratio of 1080/1480. That's "nice" but two problems. You are using To Pneuma Hagion = 1080 for the numerator but just Christos = 1480 for the denominator. That gives an uncomfortable tension of a ratio between an arthrous title to an anarthrous title.

Hmm. 'Christ' is normally used without the definite article and I'm assuming that the Greek Christos is the same. I don't see a great tension there, although it would have been preferable for it to have been 2368.

Incidentally, are you aware that the sum of G1.1, divided by the product of the word values in G1.1 gives 888? A scaling factor of 10^17 is required.

(2701/(913x203x86x401x395x407x296)) x 10^17 = 888.0383. . .

I haven't seen it anywhere, though it's hard to believe no-one has found it before me.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pmAlso, the reduced fraction is 1080/1480 = 27/37 which are two genset numbers. So the real significance comes from the genset which is derived from a coherent analysis of Genesis and John and integrates with the continued fraction of inverse alpha, not by comparing arthrous and anarthrous titles with little motivation from Scripture. I grant that comparing Spirit to Christ (incarnation) has a nice ring to it, but it seems arbitrary and subjective, which is what I avoid in all my research.

Yes, I knew it reduced to 27/37, but forgot about your generating set.

The subject of deriving constants (mathematical and physical) from G1.1 (I've found several) should maybe be the subject of another thread. I would argue that they are justified and I've found them from geometric constructions, particularly gamma (Euler's constant) and c (the speed of light). But this discussion is beginning to branch off, so I'll leave it for now.

RAMcGough wrote: Sat Feb 28, 2026 12:33 pmPlease forgive me brother if my words sound sharp! I'm just speaking from the heart with my mind on fire!

No offense taken. In turn I would ask you to forgive me for any offense I may have caused you in discussions we've had over the years. I enjoy these discussions too and I always learn something from you.