Normally, I don't respond with videos when
I'm pestered by trusting individuals who have
obviously been duped by snake oil salesmen,
but this one was just too good to pass up.
YouTube user C No-one suggested that I checkout
an article on the Veteran's Today website.[1]
The article was posted, back in February of
2015, by the well-known 9/11 truth-seeker,
Dr. Kevin James Barrett.[2]
Barrett, however, didn't write the article,
just the forward.
The meat of the article was written by Mark
Wollum, with help from his dead brother Scott
who passed ideas to Mark in his dreams.
No, really.
The premise of his article (or their article)
is that the value of pi that we've been using
for over 2000 years is, well... wrong and
that the true value, which has also been known
for over 2000 years, has been a closely guarded
secret of the illuminati (or some other group
hell-bent on world domination) for as long
as we've known the true value of pi, over
2000 years.
And, the reason for this duplicity is to prevent
us from developing economic alternate energy
sources.
Yes, I think we have all the conspiracy and
paranormal bases covered here.
Wollum's work exploits the well documented
coincidence that pi is approximately equal
to 4 divided by the square root of phi, the
golden ratio.[3]
What's the golden ratio?
Well, if the ratio of two line segments, the
longer segment over the shorter segment, is
equal to the ratio of their sum over the longer
segment, then that ratio is called the golden
ratio.[4]
In his rigorous mathematical treatise, Wollum
starts with a circle, then he adds a square
with the same center as the circle.
The diameter of the circle is 4, so its circumference
is 4 pi.
Each side of the square has a length of pi,
so its perimeter is also 4 pi.
Of course, this diagram cannot be constructed
with a straight edge, compass, and pencil,
so Wollum isn't presenting a geometric proof.
His approach is more along the lines of a
"What if..." analysis.
Wollum refers to this "same perimeter" relationship
between his square and circle loosely as "squaring
the circle," but of course, for us purists
"squaring the circle" is the classic problem
of constructing a square and a circle with
the same areas, not the same perimeters.
To make the math easier, Wollum doubles the
dimensions of his square and circle.
He then inscribes a right triangle with hypotenuse
4 and height pi.
Then, he adds another right triangle with
height 4 and width pi.
Of course, the circle and square are just
window dressing.
You don't need them to draw the two right
triangles, which also, incidentally, cannot
be constructed with a straight edge, compass,
and pencil.
So, again, we are not dealing with a formal
geometric proof here.
Now, a question for the casual student...
Are these two triangles similar?
Well if you Google "similar triangles" you
find that two triangles are similar if their
corresponding angles are congruent (meaning
equal).
And, as a consequence of having equal angles,
their corresponding sides will be proportional,
meaning the ratio of the corresponding sides
will be equal.
So, first off, are the two corresponding angles,
which I've labeled here as theta and omega
(not to be confused with a popular fraternity),
equal.
Well, we could use a couple of trig identities
to figure that out.
Google tells us that the cosine of an angle
is the length of the adjacent side over the
hypotenuse.
So, for the left triangle, we find the cosine
is pi over 4 and the angle is something close
to 38 and a quarter degrees.
Google also tells us that the tangent of an
angle is the length of the opposite side over
the adjacent side.
So, for the right triangle, or the one on
the right that is, the tangent of omega is
pi over 4, which is an interesting coincidence,
and the angle is something close to 38 and
an eighth degrees.
Wow!
The error between the two angles is only about
a quarter of a percent.
Close.
Really close, but not equal.
So, these two triangles are not similar triangles.
But, what if we didn't know anything about
trig?
Or, what if we thought that trig didn't work
because the value of pi is wrong?
Well, we could just use the fact that if these
triangles were similar the ratios of all three
pairs of corresponding sides would also be
equal.
But, before we can check that, we need to
find the length of the missing sides.
Most everyone knows the Pythagorean Theorem
by heart, but if you don't, just ask Google
and you'll discover that the square of the
hypotenuse of a right triangle is equal to
the sum of the squares of the other two sides.
So, armed with this powerful formula, we can
calculate the exact values and approximations
of the lengths of the missing sides.
Thanks, Google!
Now, let's label the angles of the two triangles
and find the ratios of the corresponding sides.
Again, we get numbers that are close, all
in the vicinity of 1.27, but no two ratios
are equal.
So, even though the sides of these two triangles
are nearly proportional, the triangles themselves
are not similar.
Wollum also writes about the amazing Kepler
triangle[3] and what he considers the most
important property of the Kepler triangle,
which is that the hypotenuse times the short
side equals the long side squared.
Uh, yeah...
Of course, you shouldn't use this, um... let's
call it a "side length test," to test to see
if you have a Kepler triangle, since it's
not part of the definition of a Kepler triangle.
This relationship is merely a coincidence.
The only way you can truly tell if you have
a Kepler triangle on your hands is by the
ratio of its sides.
The hypotenuse over the short side is phi,
the golden ratio, and the long side over the
short side is the square root of phi.
By the way, a Kepler triangle can be constructed
with a straight edge, compass, and pencil.[3]
Just throwing that out there.
So, are either of these two triangles Kepler
triangles?
Well, if we calculate the ratios, we see that
none of the ratios for either triangle come
out to values required for a Kepler triangle.
Again, they're very close, less than a quarter
of a percent error for the triangle ABC and
less than one tenth of a percent for triangle
DEF, but still not close enough to call the
election a tie.
Therefore, neither triangle is a Kepler triangle,
nor are the two triangles similar triangles.
But, why let facts like that get in the way
of a good story?
After setting the stage with all this wonderful
information about right triangles, Wollum
inscribes a couple of Kepler triangles within
circles.
One triangle has a long side of 4, while the
other has a short side of pi.
Then, he pulls the ol' bait and switch by
inscribing the two triangles we've been studying
inside circles.
Does drawing a circle around a right triangle
make it a Kepler triangle?
'Course not.
Inscribing a triangle inside a circle so that
its hypotenuse is the diameter of the circle,
simply means you have a right triangle, inscribed
in a circle.
That's it.
Nothing more.
So then, Wollum applies his "side length test"
to the two subject triangles and discovers
that by factoring everything over to the left
side, he gets the exact same expression for
the two triangles.
Wow!
How can you argue with that?
I mean, this is significant.
Right?
I mean, obviously this makes the two triangles,
um... something.
Right?
I mean, there has to be some significance
to all this.
Right?
So, what does arriving at the "same expression"
have to do with the value of pi?
Nothing, really.
We already know that the two subject triangles
are not similar, nor are they Kepler-ian (maybe).
So, what happens if we bench pi and replace
him with his slightly shorter brother 3 in
these two triangles?
Well, we get the same expressions for both
triangles again.
Holy hoodwink, Batman!
And, what if we swap out both pi and 4 for,
let's say, 5 and 6?
OMG!
We get the same expressions again!
How can this be!?
Well, the fact is that anytime you have two
right triangles and the cosine of one triangle's
acute angle equals the tangent of the other
triangle's corresponding acute angle, then
Wollum's "side length test" is going to give
you the exact same expressions for both triangles.
Regardless.
Arriving at the "same expressions" is just
a coincidence.
The Kepler triangle happens to be a special
case where the sine and tangent of corresponding
acute angles are equal because of the geometric
progression of the length of the sides.
So, Wollum committed what amounts to a logical
fallacy, which happens when you don't mind
your P's and Q's.
So, what about our subject triangles?
Is the derived "same expression" an equation?
No.
Pi to the fourth power plus 16 pi squared
minus 16 squared is about minus 0.677,
not zero.
If we substitute the variable x for pi, we
get a quartic equation which is easily solvable
by substitution.
The positive real root is the square root
of the product 8 times the sum, square root
of 5 minus 1.
I'll leave it as an exercise for the casual
student to prove that this expression is equal
to 4 over the square root of the golden ratio.
But, bottom-line for Wollum is that if pi
were the magic number, 3.1446-something, then
this expression would be a true equation.
But, it's not.
Of course, if 3 in the 3-4 triangle were 3.1446-something,
it would be a true equation.
And, if 5 in the 5-6 triangle was about 4.72,
it too would be a true equation.
Wollum's logic doesn't work any better for
deriving the true value of pi than it does
for finding the true values of 3 or 5.
But wait!
There's more...
In another exercise, Wollum multiplies each
side of the triangle with pi on the long side,
by three different values to derive the triangle
with pi on the short side.
Of course, if these were similar triangles,
then their corresponding sides would be proportional
and the values, X1, X2, and X3 would be equal.
Instead, Wollum comes up with three different
values for X1, X2, and X3, each close to 1.27.
Then, using his trusty "side length test"
he derives an entirely new expression, which
can actually be factored into the previous
expression times a quartic equation with no
real roots, which is kinda cool really.
Replacing pi with the variable x, we might
be able to solve this octic equation by substitution,
making a quartic equation, but instead I got
lazy and decided to use an online polynomial
calculator and discovered that the two real
roots of this equation are identical to the
real roots of the previous equation.
It was at this point I realized that Wollum
(either Mark or his dead brother Scott, who
talks to Mark in his dreams) is brilliant.
No, seriously.
A quartic equation and an octic equation both
with the same real roots.
This is the kind of crap I used to pull off
in high school.
It drove my math teachers up the fricken wall,
especially Mr. Spruell.
Oh God, that man hated me because I would
invariably use "alternate" methods to solve
problems that gave the correct answer (or
nearly the correct answer as Wollum is doing),
since the book methods were always so pedestrian.
Anyway...
Whether he's trying to bemuse the mathematically
illiterate or simply amuse those of us who
see through the clutter, he obviously understands
what he's doing and is intentionally leading
his readers down the rabbit hole.
I don't need to continue and spoil the rest
of Wollum's fanciful narrative for you.
But, I encourage you to check out both of
his articles and see if you can find the ever-so-slight
flaws for yourself.[1, 5]
Or not.
But, as a final word, I just want to note
how humorous it is when people with a conspiratorial
mentality latch on to articles like this and
accept them verbatim, without question, because
it supports their misguided view of how the
world works.
Believing there's a sinister superpower controlling
everything is more comforting to them than
accepting the random coincidences in their
lives.
For the most part, I don't think these people
intentionally refuse to understand what might
be considered the unacceptable aspects of
reality.
They're not stupid.
They're just blind to anything that could
shatter their world, just like Bernard in
Westworld was blind to his own design plans.
I actually feel sorry for people like C No-one,
who can react to something as simple and elegant
as Archimedes' method of deriving limits for
pi, by echoing Bernard's, "It doesn't look
like anything to me."
It might be fun to explore the conspiratorial
mentality and how it seems to be associated
with an acceptance of the paranormal in a
later video, but for now, it's past my bedtime
and I think my dead older sister really needs
to talk to me.
Good night, all.
