Talk:Pascal's pyramid
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Why is this called a pyramid? The base is a triangle, so isn't it a tetrahedron?
- A tetrahedron is a pyramid. Factitious 00:47, Jun 25, 2005 (UTC)
- @Factitious: A tetrahedron is a type or pyramid. And actually it is known as Pascal's triangle, hence I have suggested a merge. ∞😃 Target360YT 😃∞ (talk · contribs) 06:03, 10 October 2016 (UTC)
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> Summing the numbers in each column of a layer of Pascal's pyramid gives the nth power of 111 in base infinity (i.e. without carrying over during multiplication), where n is the layer - 1.
Is it really accurate to call this "base infinity"? It's more accurately base 10 (but without said carry-over). Actually, more generally, it's the nth power of 111 in any base. For example, in base 9, 111^2 = 12321:
1 + 2*9 + 3*9^2 + 2*9^3 + 1*9^4 = 8281 = (1 + 9 + 81)^2.
--Matthew0028 13:53, 14 February 2006 (UTC)
- Yes, base 9 works for 111^2, however as you get into higher powers of 111, you are forced, according to the rules of multiplication, to carry over. Higher number bases allow for higher nmbers to not be carried over. Base "infinity" makes the rule applicable to all powers of 111. I take the blame for making up the term "base infinity", however in this situation I feel that it is quite a good description of what is actually happening.
- What is actually happening is a degeneration, a summing of digits of partial results of the power. You can easily avoid this, even without the "base infinity", just insert some zeroes:
1 1 40 400 1000 100000 16000 16000000 190000 1900000000 1600000 160000000000 10000000 10000000000000 40000000 400000000000000 100000000 10000000000000000 --------- ----------------- 111^4 = 151807041 10101^4 = 10410161916100401, or 1 04 10 16 19 16 10 04 01
- which is the layer columns sum you were looking for. But again, this is just another degeneration of the original layer itself. Let's insert some more zeroes:
10000000101^4 = 10000000404000006120600041212040104060401, or 1 00 00 00 04 04 00 00 06 12 06 00 04 12 12 04 01 04 06 04 01
- and you get the original layer itself. There's a formula for inserting the zeroes in Pascal's simplex.
- (endless.oblivion (talk) 00:24, 4 April 2010 (UTC))