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.

In Other Words

RANT#20: S.1: Now the real object of true naturalists ... (is) but to find out the connection of known phenomena, and by deductive reasoning, to obtain a knowledge of hithterto unknown phenomena. ...
S.4: I had occasion, just lately,to use the word "naturalist." ... It must, however, be noted that the naturalist, as at present generaly understood, is a student of living nature only. He has certainly no exclusive right to so excellent a name. On the other hand, the physicist is a student of inanimate nature, in the main, so that he has no exclusive right to the name, either. Both are naturalists. ... Then about the other set of men. Are they not essentially students of the properties of matter, and therefore materialists. That "materialist is the right name is obvious at a glance. ... For my part I always admired the old-fashioned term "natural philosopher." It was so dignified, and raised up visions of the portraits of ... usually highly respectable-looking elderly gentlemen, with very large bald heads and much wrapped up about the throats, sitting in their studies pondering calmly over the secrets of nature revealed to them by their experiemnts. ...
S.7: I shall, therefore, in the next place make a few remarks upon mathematical investigatons in general, a subject upon which there are many popular delusions current, even amongst people who, one would think, should know better. ...
S.8:There are men of a certain type of mind who are never wearied with gibing at mathematics, at mathematicians, and at mathematical methods of inquiry. ... Plainly, then, the anti-mathematician must belong to the same class of the paradoxer, whose characteristic is to be wise in his ignorance, whereas the really wise man is ignorant in his wisdom. ... What is of greater importance is that the anti-mathematicians sometimes do a deal of mischief. For there are many of a neutral frame of mind, little acquainted themselves with mathematical methods, who are sufficiently impressible to be easily taken in by the gibers and to be prejudiced thereby; and, should they possess some mathematical bent, they may be hindered by their prejudice from giving it fair development. We cannot all be Newtons or Laplaces, but that there is an immense amount of moderate mathematical talent lying latent in the average man I regard as a fact; and even the moderate development implied in a working knowldge of simple algebraical equations can, with common-sense to assist, be not the only the means of valuable mental discipline, but even be of commercial importance (which goes a long way with some people), ...
S.9:"Mathematics is gibberish." Little need be said about this statement. It is only worthy of the utterly iliterate. "What is the use of it?" ... Now, similar remarks to thse I have often heard from fairly intelligent and educated people. They don't see the use of it, that is plain. That is nothing; what is to the point is that they conclude that it is of no use.... But what is the use of it, then? Well, it is quite certain that if a person has no mathematical talent whatever he had really better be doing something "useful", that is to say, something else than mathematics, (inventing a dynamo, for instance), ... This is quite a personal question. Every mind should receive fair development (in good directions) for what it is capable of doing fairly well. People who do not cultivate their minds have no conception of what they lose. They become mere eating and drinking and money-grabbing machines. And yet they seem happy! There is some merciful dispensation at work, no doubt....
S.10:Mathematical reasoning is, fundamentally, not different from reasoning in general. And as by the exercise of the reason discoveries can be made, why not by mathematical reasoning? Whatever were Newton and the long array of mathematical materialists who followed him doing all the time? Making discoveries, of course, largely assisted by their mathematics.I say nothing of the pure mathematicians. Their discoveries are extensions of the field of mathematics itself - a perfectly limitless field. I refer only to students of Nature on its material side, who have employed mathematics expressly for the purpose of making discoveries....
S.11:Mathematics is reasoning about quantities. ... If there be something which cannot be reduced to a quantity...then that something cannot be accurately reasoned about, because it is in part unknown. ... The unknown is not necessarily unknowable; ... But there must be an ultimate limit, because we are a part of Nature, and cannot go beyond it. Beyond this limit, the Unknown becomes the Unknowable, which it is of little service to discuss, though it will always be a favourite subject of speculation. ... The assumption of a special act of creation, either now or at any time is merely a confession of ignorance. We have no evidence of any such disconinutities. ...
S.12:It is exceedingly remarkable that the scientific spirit (asking how it is done), which is so active and widespread at the present day, should be of such recent orign. With few exceptions, it hardly existed amongst the Ancients (who would be more appropriately termed the Youngsters). Now, in the development of our knowledge of the workings of Nature out of the tremondously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in some respects a very limited, in others a very important part. ... facts are of not use, considered as facts. They bewilder by their number and their apparent incoherency. ... Theory is the essence of facts. Without theory scientific knowledge would be only worthy of the madhouse. ...
S.14:It will have been observed that I have said next to nothing upon the study of pure mathematics; this is a matter with which we are not concerned. But that I have somewhat dilated (and I do not think needlessly) upon the adantages attending the use of mathematical methods by the materials to assist him in his study of the laws governing the material universe, by the proper co-ordination of known and the discovery of unknown (but not unknowable) phenomena.

Clearly this is not my rant - it is however my exact feelings. This polemic was taken from the first chapter of "Electromagnetic Theory" by (maverick Englishman) Oliver Heaviside and published in 1893! He was self educated after age 16 and though he is a scientific giant as little known as his contemporary the (American Yale Professor) J. Willard Gibbs, it is they who are credited with establishing the modern form of Vector Analysis in their real work of phenomenal (cis?) physics. It should be noted that the emphasis throughout is mine.

Picture Lessons

Rant#19: My drawings may lack a little. The scales in the drawing are supposed to be unbalanced with the coins up and the empty pan down and the purse empty implying that sometimes you give your all and it's still not enough. It's the logic condition of necessary and sufficient for proof. The cornucopia is the Horn of Plenty - I wish I could say the quote is original. How's that for a rant!

Forward to the Past

RANT#18: Harkening back to Rants 4, 5 and 6 when I complained about being ignored by the ortho-physics community over their lack of interest in my findings on higher dimensional Argand diagrams and how I couldn't find anything about it on the Web, well here we go again on all three counts. There's good news and bad news. I pulled a 1985 Dover edition of a 1964 title "A history of Vector Analysis" and started re-reading it and came to realize a lot of my thinking had been influenced by it. It's author is Michael J. Crowe, now an Emertius Professor of History of Science at Notre Dame. Since he published just a year after I got my PhD I figured he was a lot older than I but to my surprise I Goggled him to find he's a year younger. The good news is that I highly recommend the book if you want to see the face of the human side of math rather than the sterilized presentation of the subject. The bad news is that I thought I could catch his interest with an email about my vector "discovery" only to be once again unacknowledged. As for the "discovery" (which I still believe is original) it turns out that the best reference to higher dimensional "Argand" diagrams followed Monsieur Argand (1806)by the Irishman Sir Willian Rowan Hamilton of Quantum Mechanical operator fame. His discovery (invention?) of a early variety of vectors he called Quaternions was driven by a passion for understanding the imaginary quantity (square root of minus one). So he developed (c. 1848) a four component real number quantity consisting of a scalar and three component vector with multipliers labeled i, j, and k whose properties were that their squares were equal to minus one. In my Phenomenal Universe paper I borrowed Hamilton's quaternion by suggesting that the scalar was a radial measure of a set of real x, y, and z axes (like the Pythagorian theorem with r^2 = x^2 + y^2 + z^2. Not that the quaternion isn't curious enough but a quote by Hamilton is doubly perplexing since my thrust is that my three dimension Argand diagram collapses into 4 dimensional spacetime (with an added parallel imaginary dimension to the negative time axis. Anyway the quote is: "Time is said to have only one dimension, and space to have three dimensions. […] The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." — William Rowan Hamilton (Quoted in R.P. Graves, "Life of Sir William Rowan Hamilton"). How he jumped from one real and three virtual to three real and one virtual leaves me perplexed - but there you have it.

Planetheon Video

RANT#17: Spent the month of August finishing the Folly dome mural. You can also find it on YouTube at VAVINIVILLA. It's shamelessly copied from a ceiling fresco of the planetary gods by Veronese (Paolo Caliari) in the 16th century Palliadian Villa at Maser Italy. The 2" tiles along the bottom edge are four quatrains from Edward Fitzgerald's 5th (1887) translation of the Rubiayat of Omar Khayyam:
Oh Thou, who Man of baser Earth didst make, And e'en with Paradise devise the Snake: For the the Sin wherewith the Face of Man Is blackened - Man's forgiveness give - and take! Why, if the Soul can fling the Dust aside, And naked on the Air of Heaven ride, Were't not a Shame - were't not a Shame for him In this clay carcass crippled to abide. And this I know: whether the one Ture Light Kindle to Love, or Wrath - consume me quite, One Flash of It within the Tavern caught Better than in the Temple lost outright. And that inverted Bowl we call the Sky, Whereunder crawling coop'd we live and die, Life not your hands to It for help - for It As impotently moves as you or I.
:

Growing the Universe

RANT#16: Back in Rant #12 we noted that the space around the Sun seems to be stretching at the rate of 20 cm/year/AU which converts to 0.13 kn/sec/Mpc or about 0.17% of the Hubble constant of 74.2 km/sec/Mpc. I had given 0.34 instead of 0.13 in my paper and can't find the discrepancy - either way we have a ball park figure. Now about growth rates here comes the tricky math part. If a quantity, say X, increases with time by a certain amount, say dX/dt, that does not depend on how big X is then we have an absolute growth of X+dx in the time interval dt. But if the growth amount depends on the how big X is then we have a relative growth of dx/X/dt = (1/X)dX/dt. This is a per centage growth if you muliply by 100. By itself, the ratio dX/X is a logarithmic change noted as d(log X). The log of a quantity (number) is given as the power (exponent) that a selected base number must be raised to to give that number. If we select our base as 10 then the log of 100 is 2. That is 10 to the power of 2 (squared) is 100. As odd as it may seem the log can be fractional not just whole numbers. To add to the confusion there turns out to be a "natural" base equal to 2.71828... which is also irrational like PI = 3.1415928.... The natural logarithms are written dX/X = d(ln X). We won't but it's easy to convert between ln X and log X, e.g. ln(10)=2.303 while log(10)=1. The point of all this is that the Hubble constant can be expressed in terms of a distance scaling factor that only depends on what interval of time you choose. Layzer (cited in Rant #12) using this (Friedman) constant "a" gives the Hubble constant as H = (1/a)(da/dt) and as we've seen then H = d ln(a)/dt where a = a(@t)= R(@t)/R(t=0). what we want to know is what contribution to the intergalactic Hubble constant of 74.2 would an expansion of 20 cm/year/Au contribute in our region of space. All this leads to calculating the change with H dt = d(ln a) = d ln(R/Ro) = dln R for dt = 1 year and the change in R to be from 1AU (where the rate is 20 cm/year/AU) to R the distance in AUs that light travels in one year 1. Finally we get that H = ln (R=6.35e4 AU) = 11.06! This result differs from my original paper where I had introduced a variable N that four years later has me baffled. Snce this is a Blog and not a Scientific paper I can admit mistakes. So the upshot of all this that our Sun is contributing 11.06 km/sec/Mpc to the expansion of the universe in our regions of the galaxy. And my assertion is that this is due to the radiation emanating from the Sun whose lengthening photons are causing the expansion of space about the star. I'll go further and maintain that radiation is not waves in space but waves of space.

Recovering the Organon

RANT #15: I'm afraid my train of thought might have gotten lost. Some downtime of both my health and my computer might be the culprits but the last link in my rant was that I had to discuss logatithms in order to proceed with the phenomenal expansion of the universe. And I'm just as frightened in trying to talk about math as most Americans are of hearing about math - another national disaster! Then there's the problem of not complaining enough to keep God's promise/threat at bay. So having been fairly warned of what's coming next, I thought I'd share the earliest unqualified and unquantified statements of my epihany back in 1984. As I mentioned earlier in the 77Rants I called my "theory" Corporeal Field Theory and somewhere in my epublished writings have noted that as a younger man I was neither original nor audacious enough to even consider sharing my cis-physical notions. My original proposal (unsubstantiated by any mathematical construction) I titled Corporeal Field Organon (copying Francis Bacon's(1600's) treatise on inductive scientific reasoning titled Noveau Organum, Latin for "new organized thought" or something like that), the 8-fold way (Zen) and parity on an ortho-physical Quantum Chromodynamic reference is:
1. Space flows just as inexorably as Time.
2. Space is compressed rather than curved.
3. Mass is the compression of Space.
4. Energy results as Space decompresses.
5. Entropy measures Space decompression.
6. Matter consists of standing waves of Space.
7. There is Action only with interaction.
8. Time and Space flow results from Action.
That's how it started, followed by about 15 years of reading popular scientific books and periodicals and scholarly papers when driven to and lots of math that isn't covered in the typical science curriculums. After five years of retirement I finally got the courage to face the ridicule and started to quantify my Organon with the two indisputable math findings I uncovered up to then. Since then it has all been summarily ignored. To say I don't care would definitely be sour grapes.

The Laws of Thermo

RANT #14: I don't really think that an additional term will be added to the First Law because it would never affect systems on Earth. But I do think that the source of creating new space (popcorn) ought to be considered by the ortho-phyicists. The Laws of Thermo are some of the highest achievements of phenomenal physics. The First we already cited as the conservation of energy: in any defined system of space it's changes in Energy = Work + Heat. The Second Law says the whenever Heat comes in or leaves the system there's a minimum value to the heat that's degraded by the flow called the reversible Entropy = reversible heat / temperature. The Third Law says that the reversible heat is an ideal that cannot be attained in reality. All real Entropy changes are greater than the minimum. It also implies that the only temperature you get the minimum is alsolute zero but that you can only get as close as you want of absolute zero but never arrive. The musical "The Wiz" had young Micheal Jackson playing the Strawman singing the line "You can't win, you can't break even, you can't get out of the game." I later learned that this is attributed to C.P.Snow - best selling physicist author of "The Two Cultures." My version isn't for gamblers but consumers: "Everything costs, more than its worth, there are no refunds." Finally about why we have a (thermodynamic) temperature scale that ranges from zero to plus infinity and is a fiction at zero: zero on the Celcius scale is -273.16 degrees. I work out a temperature scale that would track Celcius in the ambient range as follows:
A Temperature Scale that Fits the Third Law: "unattainabilty of absolute zero".
Define the PHI (logarithmic) temperature scale in terms of the ideal gas scale:
Phi = 739.9 x (ln K / ln 10) - 1797.79

Kelvin	Celsius	Phi
0.00	-273.15	-infinity
1.00	-272.15	-1797.79
10.00	-263.15	-1059.89
100.00	-173.15	-321.99
273.15	0.00	0.03
298.15	25.00	28.10
310.15	37.00	40.74
373.15	100.00	100.00
1000	726.88	415.91
10000	9726.85	1153.81
100000	99726.85	1891.71
1000000	999726.90	2629.61

Divides the liquid water range into 100 degrees; makes the cryogenic range within 2000 degrees (negative), the chemical range within 1000 degrees and the nuclear and stellar range within 2500 degrees.

Effect and Cause

RANT #13: Astrophysicist and Emeritus Professor David Layzer of Harvard wrote a Scientific American book called Constructing the Univese sometime during '80s- if you're into cosmology this is great reading. As an aside, he emailed when I sent my paper and told me that being professionally ignored is the state of things. In this book he makes the following statement: "the frequency of a photon in an expanding universe continually diminishes." I suggest the lengthening of the Wavelet (diminishing frequency) IS the cause of the currently expanding universe. Einstein showed that when the photon is subjected to gravity (which accounts for black holes) there is a so-called "red-shift" meaning the lengthening of its wavelength, recall that frequency x wavelength = the speed of light (a universal constant). then this could account for its lengthening. We are back to a phenomenal model of popcorn. The energy of a photon is fixed once its created so when it starts to lengthen it will lose height - the probability of "locating" the photon depends on the square of its height at that point along the wavelength. So the longer it travels (still in a straight line from its source) the likihood of locating it gets smaller and smaller as the crest and trough flatten. The second condition on our photon is that its energy = Planck's constant x frequency. So as our stretching photon loses its original enery where is that lost energy? Don't tell me that's now become potential energy - that's a bookkeeping fiction for stored energy. One of fundamental laws of physics is that energy is conserved. Called The First Law of Thermodynamics, it says that all forms of mechanical, gravitational and radiation (heat) energy (suppposedly in all the universe) can be converted into other forms without loss of the total amount. It's often taught that the First Law will always be correct because if another form of energy is found it will just be added as another term in the sum. If all these photons are being stretched over all this space shouldn't that be accounted for in the First Law? Perhaps it's the mysterious "dark energy" that has been postulated in the last 20 years. It might be pointed out that "space" is not just the nothing filling between two objects (across the table or the galaxy) but is populated by virtual sub-atomic particles that pop in and out of existence to give a noumenal account for real interactions. Space was first proposed as substantive bymathematician William Kingdon Clifford in 1870 but was never followed up due to his premature death.

A Serious Anomaly

RANT #12: The Astronomical Unit (AU) was established as a convenient scaling distance for planetary correlations. With the advent of satellites and laser distancing it has become one of the most precisely known distances. It is a twelve digit figure (149,597,870,700 meters) with an uncertainty of 3 in the smallest digit. An anomalous increase in the AU between Earth and Mars of 20 cm/year over the period from 1976 to the present. This is absolutely contrary to expectation. Quoting from KentuckyFC@arxivblog.com: "One explanation is that the mass of the sun is increasing (there is a mathematical relationship linking the AU to the mass of the sun. In fact, the mass of the sun ought to be decreasing because of mass loss to solar radiation and the solar wind. To explain the increase in the AU, the sun would have to be increasing it mass by 10^18 kilograms per year. That's equivalent to swallowing a good sized planetary moon or about 40,000 comets per year. Surely we'd have noticed that." The report is by John Anderson and Micheal Nieto, Anomalous Solar System Anomalies, Proc. IAU Symposium 261 or go to arXix.org and find PDF file arXiv 0907:2469. You can also see what's coming from (Ebony Dungeon) Webaddress. We know the universe is expanding and in fact increasing its rate of expansion. So what doesn't occur to these ortho-physicists that the increasing AU is related to the expansion? I have for some time suggested that Eddington's model of the surface of an expanding ballon with ink dots but in four dimensional spacetime is unnecessarily hard to fathom. It seems that the phenomenal model is a bowl of exploding popcorn with each kernal moving away from every other one. This rate of expansion is given by the Hubble constant which is assummed to be the same throughtout the universe. It has a value of 74.1 km/sec/Mpc - the Mpc is the megapasec and a very long distance 3.09e22 km. So for every unit of that distance the universe expands by 74.1 km every second. So the question is how does 20 cm/yr/AU convert to Hubble units? The answer is 0.13 km/sec/Mpc. - Break in blog - I had worked out all the conversion factors in about seven lines of tedious typing and went to save it by highlighting to copy and paste to the blog and closed the file instead. So you're spared a dubious foray. The upshot of all this is that solar radiation could account for the serious anomaly. Certainly better than feeding the sun a moon every year. The ortho-physicists won't admit a causal relationship between space and light. We'll see.

Life Assessment

Rant #11: Life Assessment
Today is my 77th birthday and I will get a little personal in assessing what I've done with it. Currently I am two weeks into a 4-6 week session of chemotherapy for acute myelocytic leukemia. So it's pretty clear I won't have too many more birthdays. My father died just after my third birthday in 1937 leaving my mother with three sons and a daughter to support. Long story short my older brother and I were placed in Girard College (a free boarding school for fatherless boys - a very good school) and my oldest brother in Hershey Industrial School (similar but a farm school). My own kids might very well be the victims of that upbringing. Growing up away from home had some tough times, both brothers and I managed to screw up by tenth grade but my last two years in high school were great with good studies and good friends. I had some trouble adapting to non-institutional life when I graduated high school but after three jobs in six months landed in a company lab that fit me perfectly - stayed for 15 months with encouragement from both my sister and all the lab staff to go to college. College was in all a great experience although not having much money had its moments. TUition was $600/year and books could be picked up with one hand and never cost more that $10. On graduating I was accepted to Penn State graduate school in chemistry. But within two weeks of starting Uncle Sam grew tired of my student deferment and drafted me. Penn State gave me a rain check. Sputnik was launched while I was in basic training so with a note from my Congressman I spent my years as a cold warrior in the Chemical Corps in Maryland measuring the solubility and viscosity of polymer and nerve gas solutions.
With a three month "early out" from the Army, I got to Penn State for 1959 Summer Session and life just got better. Studies and research were great: my master's advisor told me that I'd be studying half time, teaching half time, and doing research half time. In the fourth half I met a sociology masters candidate from Tennessee and by December we were engaged and in June 1960 married in the Chapel at Penn State. We got our Master's degrees the following January. I continued on to the PhD and she landed a great job as an assistant dean of women. By the spring of 1963 I was writing my thesis by night and doing lab checks in the day, when we had our twins (girl and boy) six weeks earlier than expected. The month of April was pretty much devoted to care and feeding and spelling each other. By Labor Day we were at Virginia Polytechnic Institute for a postion as Assistant Professor in the Chemistry Department. Barely co-ed and mostly cadets. Up to this point its been biographical but to assess this far I would say I had an untypical homelife or upbringing for the first say 20 years and for the next twenty say 1954-1974 I was pretty much at the top of my form. My wife had become a creative homemaker and a fantastic mother. We had a second son in 1966. The kids were still young enough to enjoy. I had been promoted to Associate Professor in 1968 and was due for consideration to Full Professor by the end of this period. It didn't happen. Although my "file" looked good: I had NSF grant, spent a summer at Oak Ridge National Lab with some good published results, a review chapter published in a advanced compendium that was more an evaluation of the literature than just a survey, a invited lecture at a Gorden Conference, averaged two more department committee assignments than my colleagues (the Department Head said "ask a busy man if you want to get something done.") My nomination was put forward and them abruptly withdrawn without ever any explanation. My suspicion is it was a political asassination by one or two senior faculty who didn't like my forthrightness or intolerance of phonies of we had our share. No matter. Life went on.
In 1975 I began a 15 year association with a couple of colleagues who had begun an university extension short course program teaching a hands on (laboratory) course of usually three days on Digital Electronics and on Computer Interfacing. We taught all over the country and also at the University. Our students were academics, industrial and government scientists and engineers. It was a very rewarding experience. On the personal side we bought a 100 acre farm and cropped Christmas trees (white pine) raising seedlings to six feet for three annual harvests. Trimming trees annualy was hard work but a great reprieve from the sedentary desk job and our own camp ground. In the 80's I has two books published, the second co-authored with a visiting Australian to my lab. My wife went back to both local universities to get accreditation to teach Latin. Found a job at a good suburban high school and taught 21 years to retirement. We launched the kids as reasonably well prepared in high school, good looking, and healthy. Story to follow - which is not meant to be hurtful but is part of my legacy. I say reasonably because the older son never took his studies seriously (although he showed incredible creativity all through his life) and instead became the class clown and general teacher pain. Unfortunately, his brother tended to emulate him. Unlike the father, they never straightened out in their last years. The daughter was a good hardworking and cooperative student all through school, she waited until she graduated to blow the lid off. I should finish the history before turning to my legacy. In 1988 we bought 30 acres 15 miles from our home with the intent of growing wine grapes. We needed to shorten my wife's commute of 50 miles one way. My spare time was used drasticly remodeling the interior. We sold our suburban home and eventually the tree farm. At work I facetiously titled myself Phariah Professor of Physical Chemistry and returned to teaching the undergrad physical chem lab and the digital electronics course and freshman chem whenever it fit. I was finessed out of the graduate courses in Thermodynamics and Chemical Bonding.
Now to my legacy - what do I leave behind that I produced? I see three areas: professional, personal and monumental. The latter giving a chance to end on a light note. Professionally, there are my pubications and my unpublications. Of the former it's not so much the whole works which were few enough but a theme that seemed to go through them. Most won't have a clue but I'm for reading it into the record. In my Oak Ridge paper we were able to show that the difference in the solubility of Hydrogen flouride and Deuterium (heavy hydrogen) flouride in an exotic molten salt mixture around 500 degrees completely on the difference in the rotational energy of the molecules as gases. Our next finding by my NSF graduate student had to do with solubilites of a variety of gases, helium, nitrogen and carbon dioxide, in simple molten salt solvents around 300 degrees, that the solubility of the gas depended on the area of the surface of the gas molecule being constricted from rotating and on a factor dependent on the viscosity of the solvent. Both of these results were explained by the Entropy of Solution and quite reasonable in hindsight. To make a long story short, after I retired I continued a study for the Army Chemical Center monitored by my last graduate student (and the only one who came willingly to work with me). We surveyed all the data we could find on the solubility of ammonia in various alcohols. THe data looked like a scatter gun shot - and out of the mess we were able to bring some really beautiful order based on the -guess?- the Entropy of Solution. One last piece performed by my second graduate student which remains buried in his thesis was that the crystal structure of Silver Chromate is in a metastable state when prepared at room temperature and must be heated to a higher temperature where it undergoes a so-called 2nd order transition to its stable form. That the data published of its heat capacity back in the 1930's is not correct. My unpublications all came after I retired but were based on notions I had since the '80s. I had my "flash" one day at home at lunch with the kids and my wife around the table and made some feeble notes and decided I'd call my theory Corporeal Field Theory to disinguish it from Einsteins General Field THeory. The pun is on my rank in the Army and the reference to body mass. Finally, I should admit that I had more than a little trouble recruiting graduate students to work with me, the first two were "encouraged" by the Department Head (before his betrayal) the first because I had fellowship funds and the second because one of my colleagues (and very good friend) was accumulating students at an alarming rate (he was an extrovert at the least). I had one who came wih a Master's degree and couldn't pass the prelim exams - had I really "played the game" he probably could have finished. My third student joined in the 80's and later told me I acted like I didn't want him. It took him seven years but we did so much together including teaching an electronics course at a Naval Research by driving the five hour trip or flying in a small airplane from the University airport. We have an ongoing friendship. Clearly my persona/introversion/personality offended/scared/repulsed the students who came to interview my research interests. Enough!
Briefly the twins flunked out of university. My daughter fell in with a partying fraternity clique, never got her interest in her coursework (Accounting - which fitted and still fits her personality). Her clique "boy"friend convinced her she should move away from home, so at 21 she left and then had to find a job to support her room and board. Through several jobs she ended up at a motel desk. When her married couple bosses decided on a great opportunity in Alaska she came and told us she wanted to go with her surrogate parents. They took a scenic route and when they got to Alaska she lost the job within the week. Stuck - she survived. She eventually met another underachiever whom she ultmately married - encouraged him to achieve. They have a working marriage and two really nice (and very bright) kids. Her twin brother started a year after she did. His meantime he worked as a line cook at a steak house. He had been living in "bachelor quarters." We wanted him to move back home for room and board. Both twins had keys to the house to come and go as they pleased. It lasted one year. He shone in one art course (my hope for him was architecture) his early computer skills and artistic abilities seemed a natural. He flunked out in two years. Started as a sou chef and launched his own career. Moved to Washington got a good job along with a Food Science graduate. When they broke up after several years we reconnected for about ten years, we discovered shared interests. He moved back and bought a house near us. I recently learned of a psycho study that achievement in life is associated with the ability for deferred gratification. This son would eat his Little Debbie cake before the school bus arrived. All three of our once beautiful children are now obese. Five years ago we offered $10K to get their BMI where it belonged. No takers. Finally comes our youngest. He knew he wasn't ready for university. So with our blessings he joined the Navy. When he was due for discharge, the day before we got a letter tha he was moving to Peoria with a mate where he ended up working full time night shift while she contributed little and he put them both through four years of college. When it was all over she dumped and went so far to take out a restraining order. Like his siblings, they are hard working, responsible and reliable employees. The younger son's issue seems to be choosing mates. He's back home and remarried to a wife even more obese than he. Both of his choices were intelligent and confident. He is our most independent and made it clear he'd as soon not see me than see me. Happily they still have good relations with their mother - and during my medical crisis have been there for her. I expect those who read this will wonder what kind of jerk I am - or maybe have figured it out.
The Summer before I retired I began pouring batchwise a concrete obelisk (like the Washington monument)with a 16 inch base and ultimately 12 feet tall which was laid to be the shadow caster (gnomen) of a sundial over a flat space of lawn about 20 x 40 feet. The obelisk has two hollow spaces, one near the bottom and one from the top which is formed around a 6 inch PVC pipe about four feet long. The bottom one we called a DNARIUM and holds samples of baby and adult hair from all our family (parents, three kids, two grandchildren). The top one has five brass flower cans with time capsules or each of the parents and kids. All these were sealed off by the summer of 2000. Since then the obelisks sides have been tiled with hand cast glazed tiles. The four differnt sides are dedicated to Truth, Beauty, Mystery, and Wisdom personified by Diana (air), Apollo (fire), Ceres (earth), and Bacchus (water). One column side is Shakespear's Sonnet 123 about time. This goes on and on, there are photos at my homepage Website (Ebony Dungeon) Webaddress. My second monument is a garden house (called a Folly for obvious reasons). It is octogonal concrete block (4" thick walls) structure with each wall section five feet wide and seven feet tall) topped by a seven foot radius dome made of mortared stacked bottomless beer bottles (2500 in all). The whole structure has been stuccoed and painted by the end of last summer. Last winter I prepared letter tiles to write out four quatrains of the Rubiyat of Omar Khayam. This spring and summer hope to finish a mural on the dome interior of the planetary gods: Sun, Moon, Mercury, Venus, Mars, Jupiter, Neptune, Saturn, Earth, and Uranus copied from a ceiling by Veronese at a Palladian villa at Maser, Italy. This is also one of my website pages.

Subtle Stuff

RANT #10: My sole commentator discredited my phenomenal wavelet collapse into a well-behaved sine train because the behavior of the photon is so subtle. You can get your fill at Wikipedia - there you'll find more than enough subtletries to gag a horse. There's the classical Mawell wave equations, the semi-classical, quantum mechanical, gauge field theory, on and on. Each adding tedium until you can act like a lawyer and believe anything. These are not wrong but I'm inclined to think they are different shadows in Plato's cave. There is room for simplicity and the wavelet could take up that space. Now, there are two fundmentally different types of fundmental particles. Without overdoing it, there are Fermions and Bosons. The electron is the simplest Fermion, only two electrons in a discrete system (like an atom) can have the same energy (think spacial occupation, like latitude, longitude and distance from the center) this requires three quantum numbers (n,l and m) but these two have to have a different orientation called spin (s). The spin of an electron is either +1/2 or -1/2. Feymann likened half spin to going around twice to return to where you start: a Moebius strip behaves like a spin 1/2 particle. The photon is the simplest Boson. It has no mass and a spin off 1. An infinitude of photons can occupy the same energy state. It travels in a straight line (one dimensional) and can be interpreted at as phase electric and magnetic fields perpendicular to its path. The wavelet we have seen is one of those fields so you have to imagine another sticking in and out of the plane. The energy and all other properties are defined in terms of the energy by the relation E=hν where h is a fundamental constant known as Planck's constant and ν is the frequency. The velocity is c, the speed of light - the upper limit to motion in the universe The photon has momentum associated with it although it is massless. In general a photon is created when an electron in an atom falls to a lower energy state. Once created it travels straight across the universe until it is absorbed by interacting with an atom. Most continue through space and as they do their wavelength stretches and the amplitude proportionally shrinks - the area of the crest and trough stays constant. So what has been found is a wavelet that can add on top of one in the train and double the amplitude or add to the end of the train in phase that has the properties of a Boson. The first I refer to as the wave front and the second as the wave train. Enough for now.

Camels and Comittees

RANT #9: The sorriest statement made by my sole commentator was "Gone are the days of the lonely patent clerk." This is an obvious reference to Einstein when he was working on the special theory of relativity. From what I've read he was never lonely but in fact was a self promoter - the horn of plenty being the one you blow yourself. The sorrow is in the "group think" attitude. I do not believe there was ever an idea generated by more than one person. Of course once the idea is launched it can be worked over by other minds - that's where we find that a camel is a horse put together by a committee. I'm no patent clerk just a retired professor who has tried not to be lonely by reaching out to those whom I thought would be interested in my findings. What I really thought would happen is that the professor would give one of his post-docs or graduate students the paper to criticize. The professor himself is the money wrangler managing an empire and who assesses the accomplishments of his underlings. I have come to the conclusion that these empires are counter productive generating a lot of trivial publications that shouldn't see the light of day until they can be formed into a whole. A professor should do his own research because it's an humbling experience and prevents pedantry. But these empire builders themselves are judged not by the quality of the work (there's the minimum set by the editors - tempered by the old boy's club) but by the quantity. Truth be told I would think the ideal job would be at a liberal arts college (of course every one of them now parades as a university) where the research could be pursued individually and publication could wait until it's ready. I've met more genuinely intellectual people on the faculty there than in university where they have become so specialized they can't afford the time away from "playing the game." I might as well add that I think the undergraduate students at the so-called research universities are getting screwed. Their classes if small are taught by instructors who are also in graduate school, a cut above a teaching assistant, or by those locally renowned professors in lecture halls of 500. The research university, to harken back to Eisenhower at the end of his presidency, is part of the military-industrial complex. I can remember as a sophomore (1954) when I started a subscription to Chemical and Engineering News as a student member of the American Chemical Society the big story was whether the federal government should have a hand a funding scientific research (there were pros and cons) through the National Science Fouundation - we've come a long way baby!

1776-Done John or Dungeon

RANT #8:One eleventh of the way to complete the Rants. I don't how anybody finds the blog except randomly. I tried to insert meta-names in the hope of hooking a search engine but Google said no. Anyway, the 1776 reference (besides the numerological connection to this site) refers to the song in the musical of the same name sung by John Adams: "Is anybody there? Does anybody care? Does anybody see what I see?" It seems to me that Physics today, having abandoned the phenomenal approach on the leading edge of research, instead has taken the noumenal approach of mathematics. I cite many quotes dating mostly to 1920's because that's when noumenal physics was launched but two are particularly poignant. G. N. Lewis, Head of Chemistry at Berkeley and renowned thermodynamicist, is quoted "Mathematics sometimes serves as a substitute for thought." The second one is from Alfred North Whitehead, the Oxford mathematician (and collaborator with Bertram Russell, both of whom turned to Philosophy) who later taught at Harvard: "There can be no true physical science which looks to mathematics for the provision of a conceptual model. Such a procedure is to repeat the errors of the logicians of the Middle Ages." Clearly the conflict between the phenomenal and noumenal was lost in the years that followed. It not that noumenal physics is wrong - are some amazing computational achievements. But referring again to the pre-Copernican Middle Ages, their computations using epicycles to account for the motions of the planets worked well too - however contrived they were. This is where I believe modern Physics stands. And you can believe that the Lords of Science will be quick to condemn me and my position as one more nut-case. Two more quotes from Whitehead, who as a philosopher transcended the narrowness of the scientific specialist: "During the Medieval epoch in Europe the theologians were the chief sinners in respect to dogmatic finality. During the last three centuries, their bad preeminence in the habit passed to the men of Science." "Also, the sort of person who was a scholastic doctor in a medieval university, today is a scientific professor in a modern university." So back to John Adams' questions: nobody is there - nobody cares - and I'm a peripheral nut-job whose math is too simple and not nearly subtle enough to qualify for even a glance. They are too isolated in their mutually exclusive subsets of "theoretician" and "experimentalist." So who should be there and who should care? I would think the physics graduate students and even undergrads are the likely candidates because they've learned the material of physics but not yet closed their minds.

Is This Time - Really?

RANT #7:

If you tried to really imagine Rant #6 - I might be able to help with a diagram. The top left figure illustrates how two two-dimensional Argand diagrams might be oriented to maintain their orthogonality (perpendicularity). We should note that negative space (-X, etc.) is a fiction. The origin of the coordinates can always be moved to keep the distances positive. In this figure Y is 90 degrees (PI/2) from X so we can swing the angle A through the whole plane. Since the two planes coincide along the vertical axis we know that the angles in the virtual plane must be consistent and are left with the figure in the lower left. Now plot cos A in the Real plane and sin A in the Virtual plane and add their values and you'll generate the pringle function.

Now if we go one step further and introduce a third Argand diagram of Z and vZ, we destroy the two planes but generate 3-dimensional Real space of X,Y,Z butted up against what looks like a 3-D Virtual space, BUT there's a problem. Any two vectors interact (known as the vector product) to form a third vector which is perpendicular to the plane of the two. For example, unit vectors i and j of the X and Y axes generate the unit vector k of the Z axis. So our Real space is maintained due to this interaction. The rub is when we try this in Virtual space. Because ivi times ivj generates -vk for all three possible interactions it seems that the 3-D Virtual space must collapse into a negative one dimensional imaginary axis perpendicular to the three Real axis to create 4 dimensional spacetime. Einstein and company showed early on in the 20th century that the proper distance between two spacetime points (events) is given by a Pythygorian formula of the square root of X^2+Y^2+Z^2-t^2. Thus we find that time might be created by the interactions of the three Virtual axes. As in my previous findings I have found no description of this result in the literature.

Imagine a Virtual Reality

RANT #6: What my sole commentator in Rant #5 was referring to in the second half of The Phenomenal Universe has to do with the imaginary number i equal to the square root of minus one and how functions can be represented on a graph whose axes are, say, x=real and y=imaginary=ix. These two dimensional plots are called Argand diagrams and you can Google to find all kinds of interpretations and applications. What's curious is that many of these are related to phase behavior, that is some form of time delay on the real axis since it's only the value projected on the real axis that exists. I'd run out of rant space if I dwelt on imaginary algebra so we'll borrow what's needed. What I have uncovered that I have not found anywhere in math books (or the web) is you can make three-dimensional diagrams by combining two Argand diagrams perpendicular to each other, (1) x vs. vx and (2) y vs. vy. I take the liberty of substituting "imaginary" with "virtual." Using i is a little cumberson because i, j, and k are standard notations for the x, y, and z components of a vector. When you do the real axes, x & y, will lie in a plane and the virtual axes, vx & vy, will also form a plane that is perdendicular to the real plane. Here the big deal is a function that appears in all the functions that describe atomic orbitals: it is e^imA where m is some integer (positive or negative #)(1,2,3 etc) and A is a "latitude" angle like from the North pole to the South Pole. Now we borrow from Gauss that e^#imA = cos A # isin A, read # as plus or minus. Now if you run A through 360 degrees with x to y = 90 degrees, etc you create a surface at an fixed radius from the origin that one of my students named the "pringle function" because it's the same shape as the potato chip. Enough for now,it's something to imagine and think about, I hope. Diagrams and photos are at (Ebony Dungeon) Webaddress.

IGNORE-ance

RANT #5: Abraham Lincoln said it first and then was wrong about the Gettysburg Address but I can get it right: "the world will little note nor long remember what we say here." My proof is already established. I wrote my essay "The Phenomenal Universe" and really felt it was worthy of professional criticism. First to two science magazines, American Scientist and then to AAAS Science, then to a long list of 40 university scientists, and finally to a dozen or so participants in a SPIE conference on "the nature of the photon." I sent a cover email with the paper as an attachment. I asked for criticism and got two polite declines (from the only two women on the list) and maybe a half dozen promises to look at it without ever responding again, one invitation to collaborate but no analysis of my paper and the only sincere response of all but with one of the sorriest statements: "It is fun to speculate about new fundamental theories but its nearly impossible to catch up with where the theory is now at without spending a lifetime at it. Gone are the days of the lonely patent clerk. I'm an experimentalist so no expert in any matters in the second half of your paper, which I read through. the first half I do know something about, and your treatment seems far from adding new significant elements. (You asked for an honest opinion.) Basically you discuss certain wavepacket solutions of some generic wave equation. It could be the Maxwell equations. However, this says nothing about the quantum theory of light or photons (which are not simply wavepackets, but something much deeper as illustrated by things like the bell inequality violation experiments). I have dabbled in this subject, although it pushes me beyond what I know." if you'd like the detail visit (Ebony Dungeon) Webaddress. The italics in bold are mine and represent what I call "professionalism" which is the narrowing of specialists and a future rant.

Wavelet Discovery

RANT #4: The simplest wave you can imagine would have one crest followed by a single trough with both ends extending thru all of space, say from plus infinity to minus infinity. Its wavelength can't be defined like an ordinary sine wave, say from crest to crest because there's only one crest.

Now a very interesting property of this wave, like the one shown in blue, is that if you place wavelets on either side at a distance of 3.14...(PI) times the unit length of X and add them together they form an almost perfect sine-like wavelength of 2X, that's half the unit length X is measured in. All elementary functions like sine and exponentials can be written as a sum of terms called a power series. As it turns out the series for sine(2X) and the exponential have the same terms (powers of X) in the variable X but different coefficients to those terms. It hard in this format to write out the series but we'll give it a shot by using the carat (^) to indicate an exponent: using Y so x means "times" then
Y e^(-Y^2) = Y - Y^3 + Y^5/2 - Y^7/3x2 + Y^9/4x3x2 - Y^11/5x4x3x2 + ...
and sin(2Y) = 2Y - (2^3/3x2)Y^3 + (2^5/5x4x3x2)Y^5 - 2^7/7x6x5x4x3x2)Y^7 + ...
Now if you use a calculator to (let Y=1 to keep it simple) calculate the left and right side (those demoninators of the coefficients are factorials (!) on the calculator, 5!=5x4x3x2x1), you'll find that 0.36788 = 0.3667 and 0.90929 = 0.90793 and the right sides will get closer to the left sides as more terms are added. Clearly the amplitude of the exponential is about 40% of that of the sine wave. I have found no mention of this "trick" anywhere. You'd think that if you told somebody who's interested in these things about this they'd be interested - guess again! There's more details and graphical illustration at (Ebony Dungeon) Webaddress. We'll chew on this in future blogs but enough for now, it's something to think about, I hope!

Ortho-physics

RANT #3: Ortho-physics has been overrun by likes of Heisenberg and Dirac: these are the guys who maintain that purely mathematical explanations only have to predict accurately and do not need any physical interpretation. This attitude that physical models are unnecessary and mathematical models are all that matter developed over the 1920s-'30s. What preceded this was about a century of development of physical phenomena using the strict logic of mathematics. It should be said that mathematics is not the culprit in either case. The phenomena dealt with in the 1800's included light (electromagnetism), heat (thermodynamics), solids (mechanics), liquids (fluid mechanics) and gases (statistical mechanics). This approach can be summarized as "phenomenal." The opposite approach might best be described by its antonym: noumenal. The only use of this word that I've read was by psychologist Carl Jung, who was himself an outstanding empiricist. I chose to use it in the most illustrative example with Heisenberg's noumenal "Matrix Mechanics" and Schroedinger's phenomenal "Wave Mechanics". Both approaches were ultimately shown to be equivalent in spite of the difference in their models. The last 30 years have seen the leading edge of physics devoted to the noumenal study of string theory with many of its detractors claiming that nothing predictable has been developed. I suggest that there is still a place on another edge of physics to return to the phenomenal approach. Hence we come around to cis-physics.

cis-physics

Rant #1: Many years ago, say 1984, an undergraduate major gave me a large pin which said "The more you complain the longer God let's you live." I was 50 then so I guess she knew something about me. It would be a new experience to find somebody(ies) who would not only listen to my rants but actually react, maybe even rave. A little over ten years ago I retired from teaching physical chemistry where I had had a few notions I had never seen in the textbooks or elsewhere. Among them were (1)a temperature scale that ran to infinity at both ends and paralleled Celcius in the ambient range since absolute zero is a fiction, (2)a working model of a Carnot engine by making the cylinder itself move like another piston without artifically changing the heat sinks, (3) a representation of atomic orbitals that contain an imaginary exponential term so can't be physically represented directly (i.e. without fudging), and (4) some neat graphical interpretations of Schroedinger's equation and "del-squared" which I describe as a Darth Vaderish "source of the force." Since I retired I have pursued messing around with the latter two and obtained what I thought were observations good enough to be published as ideas in physics/math/cosomology. There lies a tale that can lead my rants out of science and into areas of life, such as rants on education, social and cultural and even religious discourse if the occasion arises. Maybe God will let me live long enough to file a rant for each of the 77 years I've lived.

Rabt #2: What is cis-Physics? As a chemist I couldn't help but note some of the common position prefixes that already apply to Physics. First ones that came to mind were ortho-, meta-, and para- used on the benzene ring. Clearly metaphysics is taken and para-(normal) physics such as telekineisis, etc. is also. Since ortho- as in orthodox would seem to apply to main stream physics, I looked further to levo- and dextro- for left and right-handedness but that smacks of politics. So I'm left with cis- (near) and trans- (across or far) and picked cis- to avoid being too far out. Most of what I hope to blog I've said elsewhere but at an (Ebony Dungeon) Webaddress so strung-out that it's a wonder those who have visited made it. Of the many different "pages" on my website, this points to the one that contains two "phenomenal" math properties on space and time that I haven't been able to find elsewhere. The document is two years old and some minor revisions will be noted in future blogs. Once there feel free to roam.