A triangular number is based on the dots arranged in an equilateral triangle. An equilateral triangle with 2 dots on a side has 3 dots, 3 dots on a side has 6 dots, 4 dots on a side has 10 dots, like the ten pins in bowling.
The crude way of calculating triangular numbers is simply to add the whole numbers 1, 2, and so on until you reach the triangle root. The triangle root of 10 is 4. Add the numbers 1 to 4, 1+2+3+4, and the sum is 10.
A more sophisticated formula is square the root and then add the root. Then divide the sum by 2. (Root squared + root)/2=the triangle of the root. For example four squared is sixteen. Add four and you have twenty. Divide by 2 and you have ten. The triangle of four is ten.
This can also be written as the root plus one times the root divided by 2. ((r+1)*r)/2=the triangle of r, where r is the root.
The second is in Acts 1:15. When a replacement for Judas was chosen there were 120 people in the congregation. 120 is the triangle of 15.
The third is in Acts 27:37. There were 276 people on Paul's ship when it was wrecked on Malta. 276 is the triangle of 23.
The fourth and final one is Revelations 13:18, the infamous number of the beast 666. 666 is the triangle of 36.
It is reasonable to say this was an estimate, not an exact number, for several reasons. First, literal translations say it was an estimate. Second the boat was moving, and there were no doubt small waves on the shore, so an exact number would have been impossible to measure. Furthermore, no doubt, no one measured the distance between the boat and the shore. And finally it is a round number, round numbers are frequently, perhaps even normally estimates.
There are other verses in the New Testament where they use three digit numbers that might reasonably be called estimates. The word hundredfold is used five times, and the word hundreds is used once, Mark 6:40. The word hundreds is clearly not exact, they do not tell us how many hundreds.
The word hundredfold could be interpreted as being not exact, it is a round number and round numbers are often estimates. On the other hand if it is considered exact, it is a square number. This would give us six three digit numbers that are either squares or triangles.
But setting a hundredfold aside, at this point we can say there are five numbers that may be intended to be exact numbers and they are all triangles or squares. There maybe other numbers that I have missed. If you find them please contact me and pass on the information.
How did I get this rough estimate? I used the roots. Let's start with a simple example.
If we are looking for triangle numbers in the area of 666, then about one out of 36 numbers is a triangle. The next triangle number that is lower than 666 is 630, which is 36 less, the next triangle number that is higher is 703, which is 37 more.
So if we just had four triangular numbers we could multiply their roots together to get a ball park estimate of the probability of the four numbers all being triangles. The four triangle roots are 17, 15, 23, and 36. If we multiply the together we get 211,110. So there is would be about one chance in 200 thousand that the four numbers were all triangles, if all three digit numbers were triangles.
Once again to get a rough estimate I looked at the roots. I took the four triangle roots and the one square root and put them in a spread sheet. Then I looked at the closest square to each of the triangular numbers and the closest triangle to the square. I took the root of this number that was closest.
Then I divided 1 by each of the roots. Then I added the two fractions for each number. For example the triangle root of 666 is 36 and the closest square to 666 is 676. The square root of 676 is 26. So I divided one by both 36 and 26 and then added the two fractions together. This told me roughly what the chances were that 666 was either a square of a triangle.
Then I multiplied this fraction for each of the five three digit numbers together and the result was about one in 55 thousand. The chance of seeing this pattern is about one in a 55 thousand. That is a pretty big number.
Of course your chance of seeing an interesting pattern of some sort if you are looking for interesting patterns is much higher than that.
Still I find this quite impressive, especially when you combine it with some of the other strange things I have found.
John and Luke, however, did not write the whole of the New Testament. Other works in the New Testament were attributed to Matthew, Mark, Paul, Peter, and Jude. So we might note that any of them could have potentially messed up the pattern by including a three digit number which was not a square or a triangle. So the cooperation of these other writers might have be required to create the pattern.
Of course, the books of the Bible were not finally chosen until centuries after they were written. The writers did not know their books would be included in a Bible. Furthermore, it seems unlikely that those church leaders who chose the books knew anything about the pattern. All of this would make it difficult for early church leaders to intentionally set up something like this.
So a miraculous explination seems reasonable.
This impressed a distinguished professor at the Graduate Theological Union in Berkeley, Wilhelm Wuellner and Gerry Matatics, a friend of Scott Hahn who tried to prevent him from joining the Catholic Church and ended up joining himself.
Finally, I have a index page with links that lead to many essays providing evidence and defense for the faith.
Tell me what you think. Here is my contact information..Last Edited December 8, 2020
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