Monday, December 28, 2009
"Loki is fine and fair to see, bad by nature, very shifty in manner. He has that wisdom beyond advanced, but [it] is called sleight, and devices for all things; he would bring the Æsir even into a complete morass and often he would free them with fabrications."
"Loki er fríður og fagur sýnum, illur í skaplyndi, mjög fjölbreytinn að háttum. Hann hafði þá speki umfram aðra menn er slægð heitir og vélar til allra hluta. Hann kom ásum jafnan í fullt vandræði, og oft leysti hann þá með vélræðum." -GYLFAGINNING 33
There seems to be some wordplay going on in the Icelandic.
So, Loki was a magician and a prankster and there is an implication that he was as changable as the weather.
Sunday, December 20, 2009
These fits are just empirical and not based on any particular theory as to what is happening. The fits can be extrapolated to give an estimate of future temperatures. Note that for this plot the temperatures are given in °F while in previous blogs °C was used.
Supplemental: If σ is the standard deviation for the fit one would expect the temperatures to be within 3σ (approximately 0.5 °F) of the solid curve. These are mean values for the month so one would also have to add an additional correction to estimate bounds for the daily highs and lows for a particular location.
Friday, December 18, 2009
An analysis by month shows a similar pattern for the entire record and the best fits for each month displays a similar pattern. The following table gives the period, amplitude and phase for the fundamental frequency of the best fit. April appears to have the longest period and October has the largest amplitude. These months are mid spring and mid autumn although this might not necessarily be significant. One has to ask if these changes are due to oscillations or if they are just fluctuations in the temperatures which appear to be oscillations. To assert that these "fluctuations" are oscillations at this point would, in my opinion, be "deeming" them to be so. The cause of these fluctuations needs to be determined if it is shown that they are not just random fluctuations.
The coefficients for the quadratic portion of the fit are given below. The columns are the months, the mean values, the rates of change and half** the "accelerations" for each month. Note that on average the rates of change indicate both increasing and decreasing trends depending on the month.
*edit: The available data is from Jan, 1880 through Nov, 2009.
** 2nd edit: The numbers are just the quadratic coefficients and represent values at the beginning of the curves. One would have to set yr to 129 to estimate the 2009 values. The fit used 0 for the Dec, 2009 value which would add a slight error for the Dec curve.
Wednesday, December 16, 2009
First follow NATURE, and your Judgment frame
By her just Standard, which is still the same:
Unerring Nature, still divinely bright,
One clear, unchang'd and Universal Light,
Life, Force, and Beauty, must to all impart,
At once the Source, and End, and Test of Art
Art from that Fund each just Supply provides,
Works without Show, and without Pomp presides:
Those RULES of old discover'd, not devis'd,
Are Nature still, but Nature Methodiz'd;
Nature, like Liberty, is but restrain'd
By the same Laws which first herself ordain'd.
Of all the Causes which conspire to blind
Man's erring Judgment, and misguide the Mind,
What the weak Head with strongest Byass rules,
Is Pride, the never-failing Vice of Fools.
Whatever Nature has in Worth deny'd,
She gives in large Recruits of needful Pride;
A little Learning is a dang'rous Thing;
Drink deep, or taste not the Pierian Spring:
There shallow Draughts intoxicate the Brain,
And drinking largely sobers us again.
Fir'd at first Sight with what the Muse imparts,
In fearless Youth we tempt the Heights of Arts,
While from the bounded Level of our Mind,
Short Views we take, nor see the lengths behind,
But more advanc'd, behold with strange Surprize
New, distant Scenes of endless Science rise!
- Alexander Pope, An Essay on Criticism, 1709
Saturday, December 12, 2009
"Loki was cute and alluring* in appearance, bad by nature, very duplicitous in manner. He surpassed others in that wisdom which is called craftiness with devices for all things; he would even bring the Æsir into a complete quagmire and often he would free them with fabrications." - Gylfaginning, 33
Loki was a slippery character who did not engender trust and probably was responsible for the downfall of the gods. He had a fate similar to that of Prometheus.
(*edit: fríður og fagur can be translated as "beautiful" and "enchanting" but, allowing for changes over time, "plain" and "simple" might be a better fit. A modern translation for fríður is "peace." vélar and vélræðum → "devices" and "fabrications," at least that seems to be the impression.)
Wednesday, December 9, 2009
and said, “Assuredly, I say to you, unless you are converted and become as little children, you will by no means enter the kingdom of heaven. (Matthew 18:3)
When I was a child, I spoke as a child, I felt as a child, I thought as a child. Now that I have become a man, I have put away childish things. (1 Corinthians 13:11)
Paul may have seen naive impressionism as somewhat lacking. Perhaps this was due to his background. The ancient Greeks also may have viewed Hermes, the younger brother of Apollo, as somewhat naive.
Christ's message was one of renewal/rebirth.
Tuesday, December 8, 2009
In nature one finds a similar division between Special Relativity and General Relativity. Special Relativity is local while General Relativity deals with weaker but more "global" forces that ultimately dominate. These are the Commons and Lords of Nature.
We can probably do something similar with economics and global warming and try to get the two systems to work together better. The question is, "Are we up to the task?"
Sunday, December 6, 2009
Friday, December 4, 2009
One could view Ockham's razor as a form of naive impressionism. But it is an economy measure. One has to seek a balance between ignoring the lack of evidence to the contrary and unnecessarily complicating an explanation of the facts. Making unjustified assumptions raises doubts.
The operational rule of the scientific community appears to be naive impressionism with doubts. So we are justified in saying, "Don't trust them." As a matter of expediency, however, this may be the best way of proceeding, but under the circumstances one needs to show that the use of Relativity is justified in a particular case and that the results are reasonable. The assumptions break down if there is an asymmetry in the point of view of the observers. This may be the explanation of the imaginary values in the transformation for velocities exceeding the speed of light. Strong gravitation may bias the transformation and Einstein attempted to address this in the General Theory of Relativity. Special Relativity may still be the best first approximation to the laws of the Universe.
Thursday, December 3, 2009
The transformation is assumed to be linear and can be represented by four components of a matrix which only depend on the relative velocity.
By noting that a point in the second spacecraft doesn't move relative to itself while it appears to be moving with velocity -v to the first spacecraft, we can deduce B. The method can be generalized to find the any velocity, V', as it appears to the second spacecraft if its value for the first spacecraft, V, is known. This is the formula for the addition of velocities.
With L the same for both reference frames we can use it twice to make a transformation from the first to the second spacecraft and then back again to the first. Since we should get the original values back the result is the identity matrix. Doing the multiplications and equating terms gives two more terms of the transformation leaving only one unknown, A.
This is as far as symmetry will take us. To go further Einstein had to assume that the speed of light was a universal constant. This is not unreasonable if space is homogeneous. We just have to be careful about directions though. A ray of light moving along the common line of the spacecraft will appear to be moving in different directions to the two observers. Let's say that it moves away from the first and towards the second. This allows us to simplify the expression for C and determine an expression for A.
We then have to use the minus sign so that the direction of time will be the same in both frames of motion.
So we have found the transformation in this particular case. We can use other transformations to convert to situations that are less symmetrical.
This derivation indicates that Relativity doesn't impose any constraints on time travel. There are transformations which will convert positive changes in time to negative ones. But at the same time they will also convert positive energies into negative ones. So it seems likely that if one could travel back in time one would find oneself in an antimatter universe which would be extremely hazardous.
Saturday, October 24, 2009
63 BC Caesar elected Pontifex Maximus
49 BC Civil War
48 BC Caesar meets Cleopatra
46 BC Forum of Caesar
46 BC Julian calendar
44 BC Assassination of Julius Caesar
42 BC Apotheosis of Julius Caesar
30 BC Cleopatra's death
26 BC Alexandrian calendar
After examining this timeline one has to ask if the Julian calendar reforms were incomplete due to the assassination of Julius Caesar?
Thursday, October 22, 2009
The problem of designing a calendar with twelve month and 365 days is how to distribute the odd 5 days. The pattern seems to be consistent with using multiples of 30 7/12 rather than the more obvious 30 5/12 as seen in the calculation below. An irregularity is that the sequence is shifted by one month.
Sunday, October 4, 2009
A space elevator is a mechanism which claims to provide easy access to space. It is basically a cable with a counterweight that rotates with the Earth as it turns about its axis. The cable is heated by sunlight which varies as the elevator rotates. So there will be daily variations in the temperature of the cable also know as thermal cycling. Why study themperature variations? Most materials expand as they are heated and since the space elevator extends beyond geosynchronous orbit, 36,000 km above the Earth's surface, the change in length can be considerable.
It has been suggested that carbon nanotubes might be strong enough to create a cable that is self supporting. The carbon nanotubes are similar in structure chemically to graphite. Since the planes of carbon atoms form tubes, we would expect the density of a cable to be less than that of graphite. But one would expect the thermal properties per unit mass to be about the same.
So to approximate the thermal properties of a cable, we will assume it is made of graphite and behaves like a black body. The following calculation shows the daily temperature variation of the cable as it rotates about the Earth at the time of the equinoxes when the Sun is above the Equator. The cable absorbs energy from the sunlight which strikes it and radiates heat at a rate depending on its temperature. The method is similar to that used for simple climate models of the Earth.
The images below are a slightly condensed version of the program used to do the calculations. A simplifying assumption was that a section of the cable had a uniform temperature throughout. (For a better view of the images double click on them.)
The calculations above indicate the daily changes at the time of the equnoxes. There are two minimums because when the cable aligns with the direction of the Sun it is essentially in its own shadow and only experiences cooling. The shifts in times of the minimum and maximum temperatures for thicker cables can be attributed to thermal inertia.
The daily variations for different times of the year have to take into consideration changes in the angle of the Sun relative to the Equator. There is surprisingly little change though in the thermal cycles throughout the year. The reason is that the projection of sunlight onto the surface of the cable doesn't change that much. It is on the order of 10% as can be seen from the necessary change below. ι_S is the inclination of the rotational axis of the Earth, 23.5°. The first formula gives the declination of the Sun in terms of the angle φ which is the angle of the Sun in the ecliptic plane. θ is the angle of the cable relative to the Sun in the plane of cable's rotation.
Tuesday, September 1, 2009
In ancient times the oral transmission of information was the primary mode of instruction. There was no publishing industry and if someone wanted a text of their own they would have to make a copy of some existing text. And limits of the length of a papyrus roll and time available may have resulted in some judicious editing. Cramming may have been practiced even then.
There may not have been a organized effort to preserve and pass on knowledge in early times. What was needed tended to be passed on. What was not most likely was forgotten. The mathematical texts that we do have come from the Hyksos period when foreigners ruled Egypt. And there appears to have been an effort to recover some of the past.
Egypt may have been the victim of its own success. Its relative stability and isolation resulted in little change over long periods. The rate of change may have been too slow. Time might have passed it by and the illiterate "barbarians" were the ones who were motivated to change and ultimately ended up in control.
Monday, August 31, 2009
Sunday, August 30, 2009
Similarly, one might guess that the pyramidal numbers are proportional to the triangular numbers and divide a pyramidal number by its corresponding triangular number. One again gets a linear sequence of numbers and the factor can be easily determined.
Saturday, August 29, 2009
The sum of n 1s is n. The sum of 1 through n is n(n+1)/2. So it would seem likely that the sum we are looking for is some cubic expression and by substituting k-1 for k we get the prior sum and the difference.
We can equate the coefficients of k on both sides of the second equals sign to obtain equations for the coefficients.
Note that only unit fractions are needed for the sum. Finally we deduce the formula for the total number of blocks in a pyramid of n rows.
If we assume that the blocks are square in shape with width w and height h the volume of the pyramid is
V = n(n+1)(2n+1)w²h/6.
If we ignore the additive constants and set W = nw and H = nh we get the usual formula for the volume of a pyramid,
V = W²H/3.
A method for computing the volume of a truncated pyramid if found in the Moscow Mathematical Papyrus.
It is not necessary that all the scribes knew this rudimentary level of geometry and algebra. It would take only one master of sufficient skill to deduce the methods of calculation. The builder of the first Step Pyramid was known as Imhotep. He was immortalized by the ancient Egyptians.
It could be that the pyramids contained the mathematical knowledge of the ancient Egyptians. It is not unlikely that they would set this knowledge down in stone. At least the pyramids provided an opportunity to do so.
Tuesday, August 25, 2009
The procedure for computing the seked was to take half the base of the pyramid and divide by the height. This makes computing the slope of the Great Pyramid of Giza especially simple since the ratio of the base to height was 11/7. The seked is just 11/2 or 5½ (palms) since the unit length, the cubit, is 7 palms.
Sunday, August 23, 2009
But our main focus is the royal foot (remen) whose use can be seen in the design of the pyramids of the Old Kingdom. Let's take a look at the Great Pyramid of Giza whose original dimensions are estimated to have a base of 230.5 m and a height of 146.6 m. What does this tell us? Well, the ratio of the base to the height is 1.57231 which doesn't say much by itself but we can look at its continued fraction.
which shows that one gets a best fit for 11/7 with a scale factor of 40 cubits or 70 feet. This implies that in Khufu's time,
1 royal foot = 299.35 mm
1 royal cubit = 523.86 mm
which is very close to the values for the cubit stick from the 18th Dynasty.
An interesting coincidence is that,
11/7 = 1.5714
π/2 = 1.5708
Saturday, August 22, 2009
Another aspect of the story is the secrecy associated with the Book of Thoth. It was hidden away and protected by a series of defenses. But no matter how well something is protected the defenses can always be defeated.
Thoth appears in the story to have a dark side. While Ra, the Sun god, is the source of truth and light Thoth, the Moon, is more obscure. To obtain vengeance on the thief he first gets power over him by taking him and his family to the land of the dead. Thoth is jealous of his power and it is not easily taken.
Friday, August 21, 2009
Thursday, August 20, 2009
One of the duties of the scribes was to distribute goods. When precision was required the royal measures were beyond question.
Wednesday, August 19, 2009
Tuesday, August 18, 2009
Monday, August 17, 2009
Sunday, August 16, 2009
Saturday, August 15, 2009
The fingers of the royal cubit were marked as follows: 1finger, 2 fingers, 3 fingers, 4 fingered palm, five fingered palm, fist, and so on up to 8 fingers. The glyph for the remen is placed above the 15th finger on the scale. This appears to be what Wikipedia calls the bw (foot) which was 15 or 16 fingers. The term remen was also used to represent half the diagonal of a cubit square which was about 20 fingers. And a remen square would have a diagonal about equal to a cubit. The remen marked on the cubit seems to be associated with the forearm and not a foot. Perhaps this is an older usage that of the diagonal of a square came later.
It you look at the side of a royal cubit you will see that there are fingers which are subdivided into parts varying from 2 to 16. The Egyptian scribes needed all these divisions since they used unit fractions. But the scale shows that with 16 subdivisions they were capable of millimeter accuracy. One can compare the royal cubit with the modern engineer's scale or architect's scale.