Graham's Number
is a number named after a mathematician by the name of Ron Graham. Graham's
number is ridiculously big, and it refers to a geometry problem with respect to
higher planes of spatial dimensions. Starting with spatial dimensions, the zero
dimensional space refers to a singular point in space. The point has no
dimensions, and is a singularity. Moving up in dimensions, the first spatial
dimension refers to a connection between two zero dimension singularities via
an infinite number of points, or a line segment. The second spatial dimension
refers to connecting two lines to created a two dimensional plane. This plane
could be pictured as a square, and if all the vertices were connected with lines
including the diagonals, there would be six lines. If the lines were colored
either red or blue there are six possible configurations. In terms of binomial
expansion, there are four vertices, and two colors, so the notation would be
four choose two, which are six. That's great, but who cares? The point of the
problem is to look for certain avoidable configurations of colors and lines.
The conditions that trying to be avoided are four points, that are flat, with
six line connections, and all those line connections are the same color.
Essentially the pain tis avoid the square that was mentioned earlier being all
the same color. Obviously, this is avoidable in two dimensions, because as long
as one line is a different color, then the condition is avoidable. However, as
the number of spatial dimensions increases, the condition becomes avoidable.
The question is which spatial dimensions is the configuration avoidable. Well
spatial dimensions 2 through 12 are all avoidable, but all spatial dimensions
between 13 and Graham's number of spatial dimensions are unknown. Graham's
number of spatial dimensions makes the criteria unavoidable, but if there are
smaller numbers of spatial dimensions that are unavoidable is still
unknown.
So how big is Graham's
Number? Graham's number is best explained in a notation called arrow notation.
An example of arrow notation would be three arrow three or 3 ^ 3. This means 3
to the third power, or 27. Three double arrow (3^^3) means three arrow, three
arrow three, or 3^(3^3), that’s the same as three raised to the twenty-seven,
or 7.6 trillion. This is creating a tower of threes, or three rose to the power
three raised to the power three. Three triples arrow (3^^^3) means that there
is a tower of threes 7.6 billion threes long. The point to take away from this
is (3^3) = 27, (3^^3) = 7.6 billion, and (3^^^3) = roughly 1.26 X 10 ^
3,638,334,640,024. Meaning arrow notation gets big numbers very fast. Three
four arrow three (3^^^^3) is a ridiculously large number. Take that crazy
number, and that is how many arrow are in the next step, or (3(3^^^^3 number of
arrows) 3). Repeat this process again by taking that even bigger number and
making it the number of arrows in the next step. Repeat this process 64 times
and then you have yourself Graham's number. Unsurprisingly this number once
held the world record for the biggest mathematical number used in a proof.
Graham's number is so large that there are not names for
number to describe the number of digits in Graham's Number. Mathematicians understand enough
properties of the powers of three to figure the last 500 digits, but it is
thought that the first digit will never be known. Now how does a number like
this even get figured out? What process lead to this number? As
mentioned earlier this number has a meaning, it is the
upper bound of number of spatial dimensions that follows a
certain criteria. Ron's insight did not allow him to solve the problem,
but rather a process that would limit the bound from infinity. This
is similar to the post on the twin prime conjecture in that in both
situations the method used was not optimized, bath rather the start. However
unlike the twin prime conjecture there have not been any recent break
through limiting the bounds, but hopefully there will be.
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