RANT 27: Mysterious Circles:

It is well known that a circle is exactly circumscribed by six circles of the same diameter and that each circle exactly circumscribes a hexagon whose edge equals the radius of the circle. Why? How can the circumference which is the product of the circle diameter and an irrational number, i.e. PI, accommodate an integral number of circles? I suspect there's an easy explanation but I don't see it. Note that if you draw a circle on the circumference of the center one and then draw additional circles where the arc cuts its circumference  you obtain the six pointed "flower" (shown faintly in red). Note also that although the equilateral triangles that make up the hexagons are close packed and the hexagons are close packed like a beehive off diagonal, three circles each share on triangle that's outside.  Each circle looks like a six pointed star. If by any chance someone solves this mystery I hope they'll post a comment.

RANT 26: WRAP-UP

The Ten Axioms of Corporeal Field Theory

1. Space flows as inexorably as Time.

2. Space is compressed rather than curved.

3. Mass is the compression of Space.

4. Energy results from Space decompression.

5. Entropy measures Space decompression.

6. Matter consists of standing waves of Space.

7. Radiation consists of traveling waves of Space.

8. Space must be conserved with mass-energy.

9. Light waves are condensed photons.

10. Time is the Virtual crossproduct of Real Space.

Lighten Up

Rant 25: Now that Facebook has a billion followers it might be a good rant to quote Hyman Rickover, the father of the US Nuclear Navy. A paraphrase of his quote is: "Great minds discuss ideas, ordinary minds discuss events, small minds talk about people." To which I would add: The mindless talk only about themselves.

Now, back to ideas: It is a simple property of sinusoidal waves that they exhibit the same shape (periodicity) in both space and time. Once you can imagine how that can be expressed it becomes obvious. I have never seen it expressed except to show the same two two- dimensional graphs plotted as amplitude versus distance and amplitude versus time, that is just to change the name of the horizontal axis. But a pseudo-3D plot of these two axes perpendicular to each other, called an orthogonal projection, shows some interesting relations. We can color code the amplitude with red at the crest (maximum amplitude) and violet at the trough (minimum amplitude)using the rainbow sequence (ROYGBIV). The while lines are sine waves running parallel to each axis. The graph shows what looks like a corrugated sheet cut at 45 degrees to the bends. It also clearly shows that any point (height) on the wave moves at the velocity of the wave - which is any upward (vertical) line (same color) on the graph since distance/time=velocity.

Cast in Stone

RANT24: It's been six months since my last rant and one year since I started. The original premise was that the more you complain the longer God let's you live. My leukemia has me on a week of chemotherpy every 4 weeks. What's worse is I've run out of ideas to blog on my phenomenal cosmology gig. No matter because I'm pretty sure I'm only ranting to myself. One note, like Boltzmann's tomb, my obelisk now sports two 5.5x16" ceramic tiles of the equations of the previous two rants - cast in stone as it were.

A Simpler Spacetime Distance

RANT 23: In Rant 21 it all started out OK but got bogged down in some tedious math. It has occurred to me that there's a much straighter road to the result that is really more dramatic. The axes in Rant 21 of Imag vs Real vs Time really harken back to a similar set of axes in Rant 6 where I tried to show that the Argand axes of Re and Im can be expanded into perpendicular planes. If now we apply the cross (vector)product to Re X Im we still get the third perpendiculcar axis BUT it will still lie in the Im plane. IF that axis is in fact TIME then it must be multiplied by i, the square root of -1. Now when we consider the Pythygorian theorem in "spacetime" to obtain what is called the "proper distance" (D) which includes the time axis as well as the space axis, we would write D^2 = R^2 + (iT)^2 = R^2 - T^2. We know that R^2 = X^2 + Y^2 + Z^2 so D = Square Root of (X^2+Y^2+Z^2-T^2). This is precisely the formula used by Einstein that showed that time is not disconnected from space and that SPACETIME is a integrated concept.

One More Pearl

RANT#22: Here's a comparison of two sinusoidal wave packets. The first is a constant amplitude over seven waves and the second is a variable amplitude - these are the orange graphs. They are compared to a regular green sine wave cannot be a wave packet. Where the two coincide looks black. The fantastic feature is that these wave packets are generated from a simple exponential function that can be summed as shown over as many waves and at as many intensities as desired. If that simple function were the electric field of a photon these (orange) signals would be Bose-Einstein condensates.

What's the Time?

Rant#21:

There are two types of quantities we are interested in: scalars which have a magnitude (think point), and vectors which have a magnitude and a direction (think arrow). There are two types of multiplications (products)of vectors (say A and B): the dot(scalar product: A . B =|A||B|cos(angle between the vectors)) and cross(vector product: A x B =|A||B|sin(angle between the vectors)) products. For the perpendicular axes we will consider the angles between the vectors as 90 degrees with the sin(90)=0 and the cos(90)=1. So all scalar products will zero and the vector products will equal 1|A||B| where |A| is the length but not the direction of A only, etc.. The vectors will be unit vectors with a length equal 1, so their only difference will be their direction. These directions are alternately labeled X or i, Y or j, Z or k. There is a good dscussion of the vector product on the web at SolitaryRoad.com (who invites email contact) and once again I got summarily ignored when I emailed him/her/it with what I thought was an interesting question (see below). Now the new thing I am ranting about is combining three Argand planes as described in Rant#6. Each of these planes are X vs. vX, Y vs. vY, and Z vs. vZ. each aligned to one of the three axes: X, Y, and Z. This somehow makes for six perdendicular axes in 3-dimensional space (not something you can visualize). The figure shows the Argand planes (recall that i has two usages with some confusion: i is the label of the unit X vector and also the imaginary quantity of the square root of minus one. This is why I chose to use X as the unit vector along the X axes and vX as the imaginary axis. The vis for virtual (and looks like the square root surd) and is the same as imaginary but less confusing than i and is perpendicular to X. The direction of the vector product is perpendicular to the plane of the two vectors involved in a right-handed sense. This is shown in the figure, it would be writen X x Y = Z but if the order of multiplying were reversed the result would be negative as Y x X = -Z.
Using these rules and noting the order and sign given by the circle in the figure, i x j=k and i x k =-j, we can construct a multiplication table in the order ROW VARIABLExCOLUMN VARIABLE. There are a couple of observations we can make about the table. First off, any axis is not perpendicular to itself, it's parallel with an angle of zero, so the sin(0)=0 and the cross product is zero. This accounts for the long (main) diagonal in the table. Second, and this is the question to SolitaryRoad.com, we have no knowledge of the cross product of a real axis with its imaginary (as in the Argand plane), I postulate that it is TIME as shown in the figure and in the Table in the diagonals in the upper left and lower right quadrants. The rest of the terms follow the |A||B|sin(90) rule.

Variable	X	Y	Z	vX	vY	vZ
X	0	Z	-Y	t	vZ	-vY
Y	-Z	0	X	-vZ	t	vX
Z	Y	-X	0	vY	-vX	t
vX	-t	vZ	- vY	0	vZ	-vY
vY	-vZ	-t	vX	-vZ	0	vX
vZ	vY	-vX	-t	vY	-vX	0

This table can be simplified by treating each row as a equation where Row Variable = Sum of (Table Entry x Column Variable). If we consider the terms in the Sum as scalar products where A.B = B.A and a lot of terms will cancel in each row.

• X = 0 X + Z Y - Y Z + t vX + vZ vY - vY vZ = t vX
  
• Y = -Z X + 0 Y + X Z - vZ vX + t vY + vX vZ = t vY
  
• Z = Y X - X Y - 0 Z + vY vX - vX vY + t vZ = t vZ
  
• vX = -t X + vZ Y - vY Z + 0 vx + vZ vY - vY vZ = -t X
  
• vY = -vZ X - t Y + vX Z - vZ vX + 0 vY + vX vZ = -t Y
  
• vZ = vY X - vX Y - t Z + vY vX - vX vY - 0 vZ = -t Z
  

Variable	X	Y	Z	vX	vY	vZ
X	0	0	0	t	0	0
Y	0	0	0	0	t	0
Z	0	0	0	0	0	t
vX	-t	0	0	0	0	0
vY	0	-t	0	0	0	0
vZ	0	0	-t	0	0	0

So now there are three equations in the real coordinates (X,Y,Z) and three in the corresponding virtual coordinates all equal to our postulated variable t, time, If each of the three virtual equations is divided by the square root of -1 = v, for example:

vX = -t X becomes vX/v = -t X/v but 1/v = -v then X = t vX. In other words there are only 3 equations. Alternatively we could have started with the three equations in the real variables (X,Y,Z) and obtained the 3 equations in the form vX = -tX which is a bit more satisfying because this says that the three space (Cartesian) coordinates themselves are related to their imaginary counterparts and the negative of time. This in a sense is what Einstein established about the concept of "spacetime." That events can be separated by a Pythagorean like theorem in which the distance is given by the square root of (X^2+Y^2+Z^2-t^2) where the space coordinates are summed but the time coordinate is subtracted.