Two Transformations of Triangle 2701

[bluetriangle]

The Creation Triangle (triangle 2701, from Genesis 1.1) can undergo two transformations, both of which produce standard values of the second and third Persons of the Trinity, which is astounding confirmation of God's Triune nature and omnipotence, given that the words were penned by monotheists, several hundred years before The Incarnation and that fractal geometry was unknown until the 19th Century.

These were found by strict application of the two functions to triangle 2701. So in the case of the Sierpinski function, you cut out the largest inverted triangle from the centre that leaves the remainder in one piece, then repeat for each stage until you can't cut out an inverted triangle without leaving unconnected fragments.

[RAMcGough]

The Creation Triangle (triangle 2701, from Genesis 1.1) can undergo two transformations, both of which produce standard values of the second and third Persons of the Trinity, which is astounding confirmation of God's Triune nature and omnipotence, given that the words were penned by monotheists, several hundred years before The Incarnation and that fractal geometry was unknown until the 19th Century.

These were found by strict application of the two functions to triangle 2701. So in the case of the Sierpinski function, you cut out the largest inverted triangle from the centre that leaves the remainder in one piece, then repeat for each stage until you can't cut out an inverted triangle without leaving unconnected fragments.

Hey Bill,

Those are very interesting geometric forms. Do you have indexed formulas for them?

I see you used the anarthrous title for "only begotten son" and English gematria (with the article) for "The Holy Spirit". That's a lot of wiggle room.

Adding the article to get "the only begotten son" changes the value to 1246 and removing the article from "The Holy Spirit" changes the value to 1276. This is why I don't have any confidence in the "cherry picking" method of finding a number and then searching for phrases that match. There's no way to control for randomness. There's no way to know if it is something God intended or not. How many different gematria methods do you mix together? Six? Standard and Ordinal values for Hebrew, Greek, English?

I've wondered about titles for the Persons of the Trinity. We have God the Father (Ho Theos Ho Pater) and "The Holy Spirit" in a variety of forms that do or don't include the article. We have Pneuma Hagion = 710 = 10 x 71 (Yonah/Dove), To Pneuma Hagion = 1080, To Pneuma to Hagion = 1450 and To Pneuma tou Theou = 2200 (I always liked this one).


Years ago I noticed the God the Son (Ho Theos Ho Uios) = 1104 which struck me as potentially significant. It's not found in the Bible, but early Christian fathers writing in Greek use "Theos Ho Uios" = 1034 (only one article). I'm not sure if they ever used two articles.

Then there's another variation of ὁ υἱὸς ὁ μονογενής (ho huios ho monogenēs) or μονογενὴς υἱός (monogenēs huios) is used with and without the articles. Other titles include Son of Man. The Son of God, etc.

Given so many titles and variations it seems very hard to make a convincing case by first finding numbers through "transformations" of the text and then looking for matches.

But hey! Don't worry about me. I'm sure there's a lot going on that I know nothing about.

God bless you,

Richard

[bluetriangle]

Hey Bill,

Those are very interesting geometric forms. Do you have indexed formulas for them?

Not formulae, no, although I'd be happy to show how it works for each subtype of triangle (ie, G-triangles, G+1 triangles and G+2 triangles). I've achieved some interesting results with triangles I found in the opening verses of the NIV.

I've illustrated below the iterative process creating the Koch antisnowflake in plane geometry.

If the area of the starting triangle is A(o), then the area, A(n), of the nth iteration of the resulting antisnowflake is given by

A(n) = 1/5{2 + 3(4/9)^n]A(o)

So for n = infinity,

A(n) = 1/5{2 + 0]A(o)

= 2/5 A(o)

Antisnowflake 1489 is a third-iteration snowflake. For n = 3,

A(3) = 1/5[2 + (4/9)^3] A(o)

= 0.4176 A(o)

In plane geometry, for a triangle of area 2701 units^2 that gives an area of the third iteration of 1128 units^2. This doesn't correspond well to the digital antisnowflake, with 1489 counters, but this is partly because I could only remove one counter at each location for the third stage, to keep the figure whole.

However, in plane geometry, when n = infinity, and starting with a triangle of 2701 units^2, the anti snowflake's resulting area is 1080 units^2, which is The Holy Spirit in Greek! So there is some supporting evidence there. To summarize:

Plane geometry
Triangle area 2701 units^2
Maximum no. iterations: infinity
antisnowflake area 1080 units^2
To Hagion Pneuma/The Holy Spirit (s) = 1080

Numerical Geometry
Triangle counters = 2701
Maximum no. iterations: 3
antisnowflake counters = 1489
The Holy Spirit (s) = 1489

The area of the Sierpinski triangle, given by

A(n) = (3/4)^n

tends to zero, so the comparison with plane geometry doesn't work there.

I'm aware, though, that choosing whether or not to remove the definite article doubles the possibilities. And yes, I work with three languages and four systems, so that multiplies the possibilities by 12. That makes the identities I found for the antisnowflake and the Sierpinski triangle (with and without the definite article) 24x more likely, yes. But on the other hand these are large numbers and so we are still doing far better than chance. And 'The Holy Spirit' is far more popular than 'Paraclete', 'Comforter', 'The Spirit of God' and other synonyms. I'm not sure about the Greek, but I wouldn't be surprised if 'To Hagion Pneuma' is also preeminent.

Ordinal and reduced values are so small they can be effectively discounted for words and short names with values in the thousands, which effectively reduces the multiplier to 12.

Here, though, I am using the same system each time and it is the historically-attested method used by the ancient Greeks and Hebrews for counting and commerce, and its equivalent in modern English. If the systems were mixed I wouldn't be impressed myself.

There is also the symbolism provided by the Sierpinski triangle of the Son of God, "pierced for our transgressions" (Isaiah 53.5).

[RAMcGough]

Not formulae, no, although I'd be happy to show how it works for each subtype of triangle (ie, G-triangles, G+1 triangles and G+2 triangles). I've achieved some interesting results with triangles I found in the opening verses of the NIV.

I've illustrated below the iterative process creating the Koch antisnowflake in plane geometry.

If the area of the starting triangle is A(o), then the area, A(n), of the nth iteration of the resulting antisnowflake is given by

A(n) = 1/5{2 + 3(4/9)^n]A(o)

So for n = infinity,

A(n) = 1/5{2 + 0]A(o)

= 2/5 A(o)

It looks like you are using the formula for a continuous Sierpenski triangle. It doesn't work with the discrete triangle made with dots.

There is a formula for Sierpenski triangles give by [Number of vertices in Sierpiński triangle of order n. Here are the first entries:

3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575 ...

The formula is ST(n) = (3^n + 3)/2

It's impossible to create an actual (accurate & consistent) discrete Sierpenski triangle from T(73). You have to overlap some dots while not overlapping others in an inconsistent way. Basically, you are trying to force a pattern on a grid that does not accommodate it. This procedure only works for T(n) where n = 2^k+1.

I've attached the explanation from Grok.

image.png (217.66 KiB) Viewed 3042 times

image.png (136.86 KiB) Viewed 3042 times

[bluetriangle]

Not formulae, no, although I'd be happy to show how it works for each subtype of triangle (ie, G-triangles, G+1 triangles and G+2 triangles). I've achieved some interesting results with triangles I found in the opening verses of the NIV.

I've illustrated below the iterative process creating the Koch antisnowflake in plane geometry.

If the area of the starting triangle is A(o), then the area, A(n), of the nth iteration of the resulting antisnowflake is given by

A(n) = 1/5{2 + 3(4/9)^n]A(o)

So for n = infinity,

A(n) = 1/5{2 + 0]A(o)

= 2/5 A(o)

It looks like you are using the formula for a continuous Sierpenski triangle. It doesn't work with the discrete triangle made with dots.

Yes, I know, although the larger the triangle the more we align with the continuous figure.

The reason I showed you the formula for a continuous triangle is because the antisnowflake that results when n = infinity is exactly 2/5 the area of the starting triangle (n = 0). For a triangle of 2701 units^2 the antisnowflake has an are of 1080 units^2 (to the nearest whole number), which I thought was supporting evidence for the English 'The Holy Spirit' being by design, rather than a simple case of cherry picking.

Perhaps I need to start a thread on cherry picking, as it's something we need to thrash out. Thanks for the other information though.

Bill

[bluetriangle]

Not formulae, no, although I'd be happy to show how it works for each subtype of triangle (ie, G-triangles, G+1 triangles and G+2 triangles). I've achieved some interesting results with triangles I found in the opening verses of the NIV.

I've illustrated below the iterative process creating the Koch antisnowflake in plane geometry.

If the area of the starting triangle is A(o), then the area, A(n), of the nth iteration of the resulting antisnowflake is given by

A(n) = 1/5{2 + 3(4/9)^n]A(o)

So for n = infinity,

A(n) = 1/5{2 + 0]A(o)

= 2/5 A(o)

It looks like you are using the formula for a continuous Sierpenski triangle. It doesn't work with the discrete triangle made with dots.

Yes, I know, That was why I was referring to plane geometry, to distinguish it from numerical geometry. But the larger the pixellated triangle the more we align with the continuous figure anyway.

The reason I showed you the formula for a continuous triangle is because the antisnowflake that results when n = infinity is exactly 2/5 the area of the starting triangle (n = 0). For a triangle of 2701 units^2 the antisnowflake has an area of 1080 units^2 (to the nearest whole number), which, since it is the value of 'The Holy Spirit' in Greek, I thought was supporting evidence for the English 'The Holy Spirit' being by design, rather than a simple case of cherry picking.

Perhaps I need to start a thread on cherry picking, as it's something we need to thrash out. Thanks for the other information though.

Bill

[bluetriangle]

It's impossible to create an actual (accurate & consistent) discrete Sierpenski triangle from T(73). You have to overlap some dots while not overlapping others in an inconsistent way. Basically, you are trying to force a pattern on a grid that does not accommodate it. This procedure only works for T(n) where n = 2^k+1.

I'm sorry, Richard, but you've totally misunderstood what I've done here and how I did it. I've explained why I used the formula from plane geometry, which I hope you now appreciate, but you've also stated that I've tried to "to force a pattern on a grid that does not accommodate it." That is NOT how I did it.I forced nothing. I always create Sierpinski triangles in the same logical manner.

1. Subtract the largest inverted triangle from the centre of the starting triangle that leaves the remainder intact, rather then in fragments.
2. Repeat the same process with the three remaining triangles, and so on.
3. Stop when this can no longer be done.

So for triangle 2701 you can remove
1 x T34
3 x T16
9 x T7
27 x T4

giving a 4th-iteration Sierpinski triangle.

Using formulae and verbal descriptions may be formally accurate, but it is clumsy and unnecessary (and therefore inelegant) in this context. It's a very visual process and can be validated by simple inspection. I don't think it dishonors God to avoid formulae here. I think it makes the process more accessible. Anyone can see it is unforced and natural. It works brilliantly with all triangles and centred triangles and it has opened new fields of discovery. I've illustrated one below and can give several examples that anyone, not just the mathematically adept, can plainly see are linked to the gematria of Jesus Christ - that is, if you accept the reality of English gematria - and two beautiful Sierpinski triangles encoded in a mirrored pattern within Genesis 1.1, in the original Hebrew, 2500 years before fractals were known.

[RAMcGough]

Yes, I know, although the larger the triangle the more we align with the continuous figure.

The reason I showed you the formula for a continuous triangle is because the antisnowflake that results when n = infinity is exactly 2/5 the area of the starting triangle (n = 0). For a triangle of 2701 units^2 the antisnowflake has an are of 1080 units^2 (to the nearest whole number), which I thought was supporting evidence for the English 'The Holy Spirit' being by design, rather than a simple case of cherry picking.

Perhaps I need to start a thread on cherry picking, as it's something we need to thrash out. Thanks for the other information though.

Bill

Yes, the equations converge in the limit as n => infinity. But we are not dealing with infinity. The triangles are finite and discrete.

Do you really believe God designed His Word to fit a malformed Sierpinski triangle created with inconsistent combinations of the T(5) triangle? That really doesn't seem like His style. [Edit - I wrote this before seeing your answer. I'm sorry for the misunderstanding. I'm leaving it here with this comment so if you see it you will understand. I will now answer your long detailed explanation. Thanks for your patience.]

This brings up what I believe are two fundamental weakness in your method that I hope we can discuss as brothers in Christ.

1) Cherry Picking: This is the fundamental weakness of gematria. There will always be interesting "hits" in any random assignment of numbers to letters. How do we discern between design and chance? This problem is then amplified when we multiply the number of gematria systems and allow a global search across three languages for any words or phrases that might "fit" an whatever theme we happen to fancy. I avoid this in my work by looking for "holographic" self-reflective, recursive, fractal, coherent, precise, self-similar patterns in individual passages, like Genesis 1:1-5, John 1:1-5, and Deuteronomy 6:4-5. These are self-verifying and reveal great glory in God's Word by showing a design that is demonstrably divine. Mixing it with an undisciplined practice of gematria based on 6 methods and cherry picked phrases obscures the witness of what God has actually done in His Word.

2) Force Fitting: This is the cognitive error of imposing a preconceived pattern, template, framework, or interpretive schema onto data, observations, or phenomena, even when the fit is poor, forced, or requires significant distortion, selective emphasis, omission of inconvenient elements, or arbitrary adjustments to make everything appear to conform. It's like the game of "Four Fours" where the goal is to express every number in terms of four fours, like 1 = 4*4/4*4 and 2 = 4*4/(4+4), etc. The mere fact that we may be able to force numbers to fit symmetric shapes means nothing in and of itself, because we can do that for almost all numbers.

I've started a thread for Cherry Picking. I'll start the other one next.

Cherry Picking: A Fundamental Cognitive Error

Great chatting bro! I pray we can work together to refine our methods to be worthy of God's Glory.

Amen!

[bluetriangle]

This brings up what I believe are two fundamental weakness in your method that I hope we can discuss as brothers in Christ.

1) Cherry Picking: This is the fundamental weakness of gematria. There will always be interesting "hits" in any random assignment of numbers to letters. How do we discern between design and chance? This problem is then amplified when we multiply the number of gematria systems and allow a global search across three languages for any words or phrases that might "fit" an whatever theme we happen to fancy. I avoid this in my work by looking for "holographic" self-reflective, recursive, fractal, coherent, precise, self-similar patterns in individual passages, like Genesis 1:1-5, John 1:1-5, and Deuteronomy 6:4-5. These are self-verifying and reveal great glory in God's Word by showing a design that is demonstrably divine. Mixing it with an undisciplined practice of gematria based on 6 methods and cherry picked phrases obscures the witness of what God has actually done in His Word.

2) Force Fitting: This is the cognitive error of imposing a preconceived pattern, template, framework, or interpretive schema onto data, observations, or phenomena, even when the fit is poor, forced, or requires significant distortion, selective emphasis, omission of inconvenient elements, or arbitrary adjustments to make everything appear to conform. It's like the game of "Four Fours" where the goal is to express every number in terms of four fours, like 1 = 4*4/4*4 and 2 = 4*4/(4+4), etc. The mere fact that we may be able to force numbers to fit symmetric shapes means nothing in and of itself, because we can do that for almost all numbers.

I've started a thread for Cherry Picking. I'll start the other one next.

Cherry Picking: A Fundamental Cognitive Error

Great chatting bro! I pray we can work together to refine our methods to be worthy of God's Glory.

Amen!

Okay, over to the threads then.

[RAMcGough]

It's impossible to create an actual (accurate & consistent) discrete Sierpenski triangle from T(73). You have to overlap some dots while not overlapping others in an inconsistent way. Basically, you are trying to force a pattern on a grid that does not accommodate it. This procedure only works for T(n) where n = 2^k+1.

I'm sorry, Richard, but you've totally misunderstood what I've done here and how I did it.

And I'm sorry I misunderstood! I will now work to correct that error. Thanks for your patience!

I've explained why I used the formula from plane geometry, which I hope you now appreciate, but you've also stated that I've tried to "to force a pattern on a grid that does not accommodate it." That is NOT how I did it.I forced nothing. I always create Sierpinski triangles in the same logical manner.

1. Subtract the largest inverted triangle from the centre of the starting triangle that leaves the remainder intact, rather then in fragments.
2. Repeat the same process with the three remaining triangles, and so on.
3. Stop when this can no longer be done.

So for triangle 2701 you can remove
1 x T34
3 x T16
9 x T7
27 x T4

giving a 4th-iteration Sierpinski triangle.

Using formulae and verbal descriptions may be formally accurate, but it is clumsy and unnecessary (and therefore inelegant) in this context. It's a very visual process and can be validated by simple inspection. I don't think it dishonors God to avoid formulae here. I think it makes the process more accessible. Anyone can see it is unforced and natural. It works brilliantly with all triangles and centred triangles and it has opened new fields of discovery. I've illustrated one below and can give several examples that anyone, not just the mathematically adept, can plainly see are linked to the gematria of Jesus Christ - that is, if you accept the reality of English gematria - and two beautiful Sierpinski triangles encoded in a mirrored pattern within Genesis 1.1, in the original Hebrew, 2500 years before fractals were known.

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.

I 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.

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. It reminds me of the many people who take approximations of real or rational numbers and round them off to fit integer patterns they like. Or 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.

Here'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 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,

Richard