Natural Log E From Pascal Combinatorics

[tetrahedron]

http://www.cut-the-knot.org/arithmetic/algebra/HarlanBrothers.shtml

Dunno how new this really is, since people apparently keep reinventing the wheel in mathematics anyway.

 

Jess Tauber

[tetrahedron]

I started trying this procedure on the Pascal shallow diagonals (the only other straight-line relation that gives finite products). An interesting pattern is popping out.

 

Rows as you must already be aware sum as units to successive powers of 2 (obviously with inter-row ratio 2). When you do this for shallow diagonals you get Fibonacci numbers, with their inter-diagonal ratio of limit Phi at infinity.

 

In my examination of decimalized expansions of the straight-line relationships in the Pascal system I note the well-known fact that the inter-row ratio was 1/11 (individual row sums here being successive powers of 11), while that for the edge-parallel diagonals is 1/9 (their individual values being successive powers of 9).

 

Interestingly the ratio for the decimalized expansions of the shallow diagonals is 1/(5+sqrt35), a bit shy of 1/11 (where we use sqrt36 instead), and that of the columns is 1/(5+sqrt15), a bit shy of 1/9 (where we use sqrt16).

 

So now we run the procedure that gives e as infinity limit for rows on the shallow diagonals. Two interlacing patterns show up. I haven't yet worked out the dynamic of one, but for the OTHER the ratio between EVERY OTHER shallow diagonal is always a successive power of 2! It will be a shock if the limiting ratios of the two patterns turn out to be known numbers.

 

We can relate the Pascal system to e, and to Phi. External to the Triangle we can link them to Pi. What else is in here?

 

Jess Tauber

Edited September 5, 2013 by pascal

[Turtle]

snip...

 

So now we run the procedure that gives e as infinity limit for rows on the shallow diagonals. Two interlacing patterns show up. I haven't yet worked out the dynamic of one, but for the OTHER the ratio between EVERY OTHER shallow diagonal is always a successive power of 2!

 

We can relate the Pascal system to e, and to Phi. External to the Triangle we can link them to Pi. What else is in here?

 

Jess Tauber

 

As usual I'm not following everything you see, but a particular phrase has tickled my tether. Not sure if by 'successive' you mean that all powers of 2 occur as you skip diagonals, but I have conjectured that all 2^Prime are non-polygonal. Me and my collaborators haven't mustered a proof, but it's been checked out to 2¹⁰⁰⁹ if I recall correctly and the conjecture holds. Well, that's all I got for what it's worth Jess. :shrug:

[tetrahedron]

Yes ALL the powers of 2 occur between these every other shallow diagonal after we run the 'e' procedure on them. So the question I have is whether this is related to the row sums, where as units the sums are successive powers of 2.

 

Jess

[tetrahedron]

I can't be sure- just went to the 30th Pascal shallow diagonal, but it looks like the ratios, at infinity, for the two different alternating patterns are 1/sqrt2e vs. 2sqrt2e. The latter might be the basis of all those powers of 2?

 

Hmm- just realized if we invert the first ratio and multiply by the second we should get limit 4e.

 

Jess Tauber

Edited September 5, 2013 by pascal

[tetrahedron]

Well I was mistaken- the two resulting series of number patterns don't relate, on the surface anyway, to e in any obvious fashion.

 

The smaller one's values continue to get smaller forever, and the larger one's bigger. Interestingly, however, they are coordinated. If you line the two series up against one another in just the right way, then products of the two numbers at any level are always EXACTLY 2.

 

So now we have a kind of mystery. On the one hand rows give as unitizing sums powers of two, with ratio 2 or 1/2 depending on the direction you're going, and as decimalized sums powers of 11 (note that 2=1+1, relating the two sets). Brothers' procedure yields limit e at infinity.

 

Shallow diagonals as unitizing sums yield Fibonacci numbers- their limit ratio at infinity of course is the higher or lower value of Phi depending on the direction you're going. The ratio of decimalized sums is 1/(5+sqrt35) or inverse.

 

Powers of 11 as in the row decimalizations, are in one directio n giving ratios 1/11 between rows, which can be restated as 1/(5+sqrt36), thus relating the rows and shallow diagonals here. This parallels what we see in the ratios of decimalizations of the columns (1/(5+sqrt15) and those of the edge-parallel diagonals, 1/9, or restated 1/(5+sqrt16).

 

When we square the decimalization limit ratio of the columns we subtract a fixed amount from the original number, and this is identical to the fixed amount we ADD to the square of the decimalization limit ratio of the shallow diagonals. Everything is coordinated here.

 

So the question then is whether there is some functional equivalency between e on the one hand and Phi on the other, when we take stock of all the procedures used to create them from Pascal numbers? I've already been able to do this for the system of ratios of 1/9 and 1/11 with regard to the other two, distributing over sums and differences, multiples and ratios, powers and roots in complementary distribution. Heck, it may even be possible that logs and natural logs are similarly differentiated in the system. Has anyone looked?

 

Jess Tauber

Edited September 8, 2013 by pascal