Sunday, April 05, 2020
Sunday, October 27, 2019
Tiger Cave and the Mahabalipuram Complex
Tiger Cave:
‘Tiger Cave’ asks you to leap out to space like a Tiger would leap on its prey, say an elephant. It says: "Leave this den and seek higher destinations". The plan is a very simple one with models of:
1. The seven planets
2. The navigational equipment
3. The space shaft
4. The space module
1. The seven planets:
The seven stone deities symbolize the seven planets. Earth is in the middle and this is like a 'you are here' marker. The planets lower below Earth are Mercury, Venus and Moon. The planets higher above are Mars, Jupiter and Saturn.
2. The navigational equipment:
The shrine has a hexadecagonal Lingam of Lord Siva with each of the sixteen sides gently curved convex.
This
must be the best preserved cast basalt Lingam of the Lord in this
part of the Universe. It could be used to calculate declination of
the sun with accuracy. It may be noted that the Lingam has the usual mark
inscribed on it.
The mark most probably stems from பரன்
உள்ளம்/paran-ullam that
morphed into frenulum in English/Latin. Only a healthy heart can make
the mark and reach the zenith. It may be noted that the mark is always 90 degrees from the
base when viewed from the top. This is explained by the space shaft that is placed next. There is also a pillar like gnomon in front.
3. Space shaft:
The space shaft is about 30 feet in height and inclined at 60 degrees, pointing East towards the Bay of Bengal.
The shaft is on the beach adjoining the sea and makes an ideal launch location being close to the equator. For all the reasons rockets are launched from Sriharikota that is about 150 KM North, this location is fit for beginning space trips. The obtuse side is 120 degrees from the ground and the frenulum is therefore always facing the worshiper to highlight this.
There ought to have been a crown or kalasam at the top that must have been dislodged by a Tsunami in the past as peg holes have been carefully sculpted on the western side of the shaft to hold one.
Apparently, inter-planetary and interstellar trips were always from near the North pole. Bogar verses that describe equatorial launches would mention கிழக்கு/East and those that describe polar launches will invariably include references to பச்சைகாரி/green clouds or the aurora in the North. The space traveler starts with learning to go to earth orbit first and then explore the planets and then the rest.
Jupiter is symbolized by the elephant and Mercury is symbolized by a lion with an elephant trunk. Sun and Jupiter are the two heavy weights of the solar system and Mercury is kept in its orbit by their gravitational pulls. The Jupiter connection with Mercury is brought home with such simple imagery. Jupiter's constant gravitational tugs could eject Mercury from the solar system. In Bogar's Karukidai Nikandu, Varahi is a reference to vai/mouth, garima/gravity/heaviness, panni/pig/boars. The Varahi rekai is part of the re-entry or exit plans for the solar system. the Varahi/bary centre for the solar system and gives the optimal trajectory to sling out of the solar system. The yali at Tiger Case seem to illustrate this principle.
There ought to have been a crown or kalasam at the top that must have been dislodged by a Tsunami in the past as peg holes have been carefully sculpted on the western side of the shaft to hold one.
The tip of the shaft has been modeled carefully. The pics below show the original and the model (east view).
Apparently, inter-planetary and interstellar trips were always from near the North pole. Bogar verses that describe equatorial launches would mention கிழக்கு/East and those that describe polar launches will invariably include references to பச்சைகாரி/green clouds or the aurora in the North. The space traveler starts with learning to go to earth orbit first and then explore the planets and then the rest.
4. The space module:
The easiest thing to miss at Tiger Cave is that the yalis are on a turtle cross section, and when partly submerged in water, its sheer poetry. Hose and elephant heads with tusks and trunk, sword handle in the overall shape of a turtle head, with port holes, observation deck (the deck with ports make it a model intergalactic space vessel) is unbelievable magic.
 |
| The turtle shaped space module with port holes and observation deck |
 |
| Turtle head details with horse, elephant heads, sword handle and port holes |
The view from the other side is even more spectacular where the turtle turns into a completely new creature with a couple of whales around it.
Mahabalipuram Complex
The entire Mahabalipuram complex again is symbolization of the Seven Worlds with Seven Pagodas and the earlier landing from Tirumala to Srisailam. There is considerable literature in verse form that supplement the site models. To give an example, the following passes as "Draupadi Ratham".
It is really a parachute model, made with bamboo and silk. Bogar verses on how to make a parachute and learn the art of gliding that would come handy during descent are as follows:
1281. பார்க்கவே வாகாஷம்பாவந்தன்னில் பரிவுடனே குதிப்பதற்கு கூண்டுசொல்வேன்
கார்க்கவே வட்டமாங்குடைதானொன்று பாங்கான குடைநிகள மகலங்கேளிர்
ஏர்க்கவே ஆறடியாவட்டவீடு யெழிலான பட்டுவடந் தன்னாற்செய்து
தீர்க்கவே பிரம்பதுவும் முப்பத்திரண்டு திறமான சக்கரமுமொன்றேமாட்டே
1282. மாட்டவே சக்கரத்திலிரும்புக்கம்பி மார்க்கமாய்த் தான்முடுக்கி வாணிமாட்டி
நீட்டமுடன் கம்பிக்குத் துணிதான்போர்த்து நெடிதான சூத்திரமாங் கயற்தான்கோர்த்து
வாட்டமுடன் தான்விரித்து குடையையேந்தி வாகாகத் தான்குதிக்கில் வாயுபூந்து
தேட்டமுடன் காற்றதுவுங் கூண்டேதூக்கும் தீவிரமாய் மனிதனுந்தான் கீழ்நோக்கலாமே
1283. நோக்கலாம் குடைதனையே கையிலேந்தி நொடிக்குள்ளே மலையைவிட்டு குதிக்கும்போது
தூக்குமே குடைதானுமனிதனைத்தான் துப்புரவாய் மனிதனங்கே துணிவுகொண்டு
தேக்குடனே பூமிதனிலிறங்கும்போது தேசமெல்லாங் கண்ணுக்கு ளணுவுபோலும்
நோக்குடனே தெரியுமென்று போகர்தானும் நேராகப் பாடிவைத்தேன் நேர்மைபாரே
1284. பாரேதானின்னமொரு சூட்சஞ்சொல்வேன் பாருலகி லிந்தவித்தைபழக்கஞ்செய்ய
சீரேதானாற்றருகே தன்னிற்சென்று சிறப்பான ஜலமதுவும் நிற்கும்போது
நீரேதான் பாளமதில் நின்றுகொண்டு நேர்த்தியாய் குடைதனையே கையிலேந்தி
தீரேதான் ஜலமதினில் குதித்தாயானால் திறமான தேகமது பழுதுறாதே
1285. பழுதுமே வாராது தேகந்தானும் பலகாலுமிப்படியே பழக்கஞ்செய்தால்
கழுதுவள தானிருக்கும் பனையின்மேலே கருவாகத்தானேறி குடையையேந்தி
தொழுமே பராபரியை மனதிலெண்ணி தொய்யாமலே பூமியிலே குதிப்பீரானால்
முழுதுமே லாசுடனே பழக்கந்தன்னால் முனையான கோபுரத்திலேறலாமே
1286. ஏறலாந் தேவதாகோபுரத்தில் யெழிலான குடைதனையே கையிலேந்தி
தேறலாஞ் சிகரபுரை மீதிருந்து தேற்றமுடன் குடைதனையே விரித்துயேந்தி
மாறலாமேலிருந்து கீழ்குதிக்கில் மதிப்புடனே தீரனாயிருந்துகொண்டு
கூறலாமிக பழக்க மதிகமாகி குன்றின் மேலேறுதற்கு குணமுண்டாமே
Bogar has summarized knowledge about equatorial launches in the (வின்) ஞான பூஜா விதி 13. The space vehicle is referred to as valai/வாலை.
While
Earth orbit trips could be made from Tiger Cave, inter-planetary and
interstellar trips were made from the North Pole, from Stonehenge and
other such sites. The reason why ley lines run from Thoothukodi to
Leh bisecting India and all the way to near the North Pole.
The Vimana at Sri Rangam is believed to have been at top of the Tirumala hills when the Nallamalai was afloat in space.
The Vimana at Sri Rangam is believed to have been at top of the Tirumala hills when the Nallamalai was afloat in space.
The entire Mahabalipuram complex is just a scale model of the earlier landing that passes as geological unconformity. The next landing is believed to be here at Mahabalipuram. The Anantasayana Perumal is believed to be the site where Lord Vishnu will appear next in Kalki Avatar. As the whole Mahabalipuram Complex is serpent shaped, Perumal here is laid directly on the rocks.
Just my take on the Tiger Cave and Mahabalipuram Complex with the hope it helps us leap into space from our backyards.
Sunday, July 21, 2019
Thursday, June 20, 2019
Cubic triples vs. Cubic quadruples
The case of the cubic triples:
Fermat’s Last Theorem states that there are no three positive integers a, b and c such thata^3 + b^3 = c^3 and more generally, there are no three positive integers a, b and c such that a^n + b^n = c^n where n > 2.
The simple proof for that is as follows.
Pythagoras theorem states:
c^2 = a^2 + b^2
If we multiply both sides by c,
c^3 = ca^2 + cb^2
Since the hypotenuse is greater than the sides, c > a and c > b
∴ ca^2 > a^3 and cb^2 > b^3
∴ ca^n > a^n and cb^n > b^n where n>2
Fermat’s Last Theorem. QED.
The case of the cubic quadruples:
In the case of the cubic quadruples, a^3 + b^3 + c^3 = d^3Take a cuboid with length l, breadth b, (diagonal of l, b is c) height h and diagonal d. It may be taken to be a cube where l = b = h. Then,
d^2 = l^2 + b^2 + h^2
Multiplying both sides by d,
d^3 = dl^2 + db^2 + dh^2
Since d > l, d > b and d > h, it follows
dl^2 > l^3, db^2 > b^3 and dh^2 > h^3
However, we can’t conclude d^3 > l^3 + b^3 + h^3 while its absolutely correct to say c^3 > a^3 + b^3. Why?
Cubic triples vs. Cubic quadruples:
The following may be noted:
[1] Non-Pythagorean integer triples may correspond to non-right triangles whose base may correspond to a cube of integer length that may be the sum of three cubes, with two cubes corresponding to the integer lengths of the other sides of such a triangle.
Non-Pythagorean integer triples d, e, f could correspond to non-right (acute or obtuse) triangles on a plane with d^2+e^2>f^2 or d^2+e^2For eg. 6^2 + 8^2 = 10^2 = 36 + 64 = 100
Also, 6^3 + 8^3 = 10^3 = 1000. 1000 does not divide into two cubes of integer lengths, but apparently, 6^2 + 8^2 > 9^2, and, 9^3 = 6^3 + 8^3 + 1^3 = 729.
In cases where d^2+e^2
The same could be scaled up, for eg. 12^3 + 16^3 + 2^3 = 18^3
It is seen that while it is impossible for a right triangle to scale up to a cube, acute or obtuse triangles may scale up to form cubes with positive integer lengths and can be divided to three cubes corresponding to the cubes of the sides of the said triangles.
[2] In the case of c^3 > a^3 + b^3, always a, b, c > 1 as otherwise one of the sides of the right triangle would be irrational vide Spiral of Theodorus and their cubes would also be correspondingly irrational. Only the Pythogorean Triples need to be considered at all as they are the only positive integers such that a^2 + b^2 = c^2. Since a * a^2 = a^3, b * b^2 = b^3 and c * c^2 = c^3, if an integer cannot even be expressed as the sum of two squares, it could not be expressed as the sum of two cubes. The smallest Pythogorean Triple is 3, 4, 5 and therefore, a, b are always greater than 2. It has already been seen that since the hypotenuse is greater than the sides, c > a and c > b, therefore ca^2 > a^3 and cb^2 > b^3.
[3] d^3 could be equal to l^3 + b^3 + h^3, as one of the three cubes could be 1, which is its own cube, and could be summed with two other cubes. For example eg. 9^3 = 6^3 + 8^3 + 1^3. It may be noted that the irrational diagonal of the square in the face of the cube gets omitted. We start with c^2 = l^2 + b^2 and since d^2 = h^2 + c^2, d^2 = l^2 + b^2 + h^2, any irrationality in the diagonal of the square face of a cube is eliminated as the internal diagonal of a cube can be expressed in terms of the length, breadth and height. While the cubic triples would never have 1 as one of the integers, cubic quadruples could have it and make a^3 + b^3 + c^3 = d^3 possible. In the case of the example, the cubes of the Pythogorean Triple 6 and 8 by themselves won’t make a cube but along with 1 cube, it adds up to 9 cube.
The cubic triples when compared with the cubic quadruples show that Fermat’s Last Theorem is accurate beyond any shadow of doubt.
Ref:
[1] https://www.calculator.net/triangle-calculator.html used to draw the figures on this page.
[2] https://plus.maths.org/content/triples-and-quadruples Triples and quadruples: from Pythagoras to Fermat by Chandrahas Halai
[3] Private email from Ravi Sundaram pointing to the Non-Pythogorean integer triples corresponding to non-right triangles.
Saturday, June 15, 2019
Wednesday, June 12, 2019
The proof for Fermat's Last Theorem
Fermat’s Last Theorem states it is impossible for a cube to be written as the
sum of two cubes, and more generally, no three positive integers a,
b, c satisfy the equation a^n + b^n = c^n for any integer value of n
greater than 2. He
remarked that the proof was too long to fit in the narrow margin of
the book he made the note. The
proof:
Pythagoras
theorem states:
c^2
= a^2 + b^2
If
we multiply both sides by c,
c^3
= ca^2 + cb^2
Since
the hypotenuse is greater than the sides, c > a and c > b
∴ ca^2
> a^3 and cb^2 > b^3
=>
ca^n > a^n and cb^n > b^n
=>
Fermat’s Last Theorem. QED.
Thursday, May 16, 2019
Geometry of circle equations in n dimensional space
The values for x, y, ... n that satisfy the circle equations translate to a single point in n dimensional space or nDspace. The geometry of the point when plotted as described earlier in the clear definition of n-dimensional spaces is as follows:
The ellipse can be plotted with the equation x^2/A^2 + y^2/B^2 = 1 and the ellipsoid with x^2/A^2 + y^2/B^2 + z^2/C^2 = 1, and n connected ellipses can be plotted with x^2/A^2 + y^2/B^2 + z^2/C^2 + ... + n^2/N^2 = 1 in the same manner.
| nDspace | Equation | Geometry |
| 1 | x^2 = 1 | Point on a number line at -1 or 1 |
| 2 | x^2 + y^2 = 1 | Point on the circumference of a circle whose radius is 1 unit |
| 3 | x^2 + y^2 + z^2 = 1 | Point on the surface of a sphere whose radius is 1 unit |
| 4 | x^2 + y^2 + z^2 + w^2 = 1 | Point on the surface of a sphere defined by x, y and z whose centre O1 is w units from the centre O. Effectively this can represent oscillation between -1 and 1 |
| 5 | x^2 + y^2 + z^2 + w^2 + v^2 = 1 | Point on the surface of a sphere defined by x, y and z whose centre O1 is w,v units from the centre O. Effectively this can represent a point on a sphere with a circular orbit |
| 6 | x^2 + y^2 + z^2 + w^2 + v^2 + u^2 = 1 | Point on the surface of a sphere defined by x, y and z whose centre O1 is defined by w,v and u from the centre O. Effectively this can represent a point on a sphere with a spherical orbit |
| 7 | x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 = 1 | Point on the surface of a sphere defined by x, y and z whose centre O2 is defined by w,v and u from the centre O1, t units from centre O. Effectively this can represent a point on a sphere with a spherical orbit t units from a third vertex |
| 8 | x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 + s^2 = 1 | Point on the surface of a sphere defined by x, y and z whose centre O2 is defined by w,v and u from the centre O1, t,s units from centre O. Effectively this can represent a point on a sphere with a spherical orbit on a given plane |
| 9 | x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 + s^2 + r^2 = 1 | Point on the surface of a sphere defined by x, y and z whose centre O2 is defined by w,v and u from the centre O1, t,s,r units from centre O. Effectively this can represent a point on a sphere with a spherical orbit t units from a third vertex to depict the position of a point in relation to three bodies |
| n | x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 + s^2 + r^2 + ... + n^2 = 1 | Point depicting relationship between n spherical bodies |
If the vertices represent the centre of gravity of bodies, it may be seen that they are also ruled by these fundamental equations. If the distance between Sun and Earth is one unit, then the distance between Earth and Moon is 0.001 units approximately, and 0.001^2 = 1 x 10^-6. It would be interesting to scale and map actual distances in the above equations for any three bodies, or n bodies in general.
It may also be noted that all function variables could be nested and graphed elegantly in nDspace.
Tuesday, April 30, 2019
Fairing Fermat's Last Theorem
Fermat
was a Superior Judge in France, who presided over several trials and
had ordered execution of many a criminal, like fake gurus of that
era, on infamous stakes. As a judge, he had to keep his distance and
in his splendid isolation, mathematics was his chief love and
companion. Nothing else comes between a proposition and its proof
that is either clearly there in one’s mind or it simply does not
exist. He enjoyed teasing and taunting his peers making outlandish
mathematical claims sans proofs, challenging them to come with one.
He wrote it is impossible to separate a cube into two cubes, and more
generally, no three positive integers a, b, c satisfy the equation
a^n + b^n = c^n for any integer value of n greater than 2. He
remarked that the proof was too long to fit in the narrow margin of
the book he made the note.
A
cube to be separated into a pair of cubes, must first be capable of
being separated into squares. The smallest Pythagorean triple is
3, 4 and 5. When squared, the numbers become 9, 16 and add to a
perfect square of 25. If they are cubed, we get, 27, 64 and 125, but
the cubes of 3 and 4 add up only to 91. If the same triple is scaled
by two, the triple 6, 8 and 10 yield perfect squares that add up,
i.e. 34 + 64 = 100, but when cubed, yield, 216 and 512 that add to
728, woefully short of 10^3. The
woeful shortage observed while cubing Pythagorean triples is because
while the diagonal of a square is square root of two, the diagonal of
the inner space of a cube is square root of three, and
the ratio
between the two
is 0.816. It
is no coincidence that 91/125 = 728/1000
= 0.728.
The ratio of the shortage is larger in the
case of
the other Pythagorean triples. When
n > 3, it
is only a
reflection
of the cube, n times in nDspace, and
therefore there is never any let in the shortage.
However,
728 is interestingly very close to 9^3 – just one
cube short of 729. Most obviously then
6^3
+ 8^3 + 1^3 = 9^3
Curiously,
729 is both a cube and a square: 729 = 9^3 = 27^2
It
is clearly possible to separate some cubes into three cubes, in the
best case with the smallest Pythagorean triple 3, 4, 5 scaled by 2.
This
could be scaled to infinity.
For
example,
12^3
+ 16^3 + 2^3 = 18^3
Numbers
like 729 and 4096, that are both cubes and squares, must be called
the Pythagorean lame doubles – lame right triangles with a hypotenuse and arm coinciding but with a missing arm, and are cubes that can be
separated into three cubes, more or less.
4096
= 64^2 = 16^3 = 8^4 = 4^6
Again,
4096 = 15^3 + 9^3 – 2^3
The
best it can be expressed in terms of cubes.
Only
lame doubles come anywhere close to aiding the separation a
cube into three smaller cubes, and separating a cube into just two cubes is simply impossible.
The
judge would hardly have asked for more proof.
Wednesday, August 23, 2017
Elliptic orbit in 5D Space
Plotting the equation X^2 + Y^2 + Z^2 + W^2 + V^2 = 1 in 5D space explains eccentricity in elliptical orbits rather elegantly:
Values of W and V can only range between 0 & 0.98 if X,Y & Z are assumed to be 0.1.
The complete chart made roughly with convert +append, -append command:
Monday, August 21, 2017
Plotting the circle equation in the 4th dimension (initial draft)
A quick summary of the calculations required to plot x^2 + y^2 + z^2 + w^2 = 1 on a graph.
First, calculate x, y, z and w using a spreadsheet, for random values that satisfy the above equation:
Plot on a graph and verify:
The figures 2 and 3 in the centre may actually explain scattering and orbital jumping. Geometrically, an infinitesimally small point P can fall only near 1 or -1 since those values alone satisfy the equation x^2 = 1, and therefore, if x^2, y^2 and z^2 are zero or close to it, w^2 = 1, and the point defined by the equation could only fall near 1 or -1 in 4D space.
Friday, July 14, 2017
A clear definition of n-dimensional spaces
- Ramanraj K
14thJuly, 2017
Introduction
2-D and 3-D co-ordinates can be plotted with absolute certainty. However, 4-D and higher dimensions in n-D spaces are not clearly defined and their co-ordinates cannot be plotted with mathematical certainty. A definition of dimensions higher than three is necessary for clarity and use in mathematics, physics and computing. This would help both man and machine to describe and visualise virtual models of the world.1-D Space
The one dimensional space is defined as a stright line along x-axis with a single vertex O in the centre.
Example: A railway track with no branches.
2-D Space
Two dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O.
Example: The screen frame, with top and left as co-ordinates.
3-D Space
Three dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O, and a third straight line z perpendicular to both x and y passing through O.
Example: A train engine, a ball, earth, sun, and moon in space.
4-D Space
Four dimensional space is the 3-D space defined by lines x, y and z as above whose vertex O1 lies on a straight line w with vertex O. Therefore, in 4-D space at least two vertices co-exist and the distance between the vertices O and O1 is defined by the 4th dimension w. The lines x and w coincide. Let a 3-D space with vertex O1 be defined as follows:
Then, the 4-D space is as follows:
In the above figure, w = 4, and the distance between the vertices O and O1 is 4 units. This is useful to define the relationship between a 3-D object with a given vertex O and another vertex O1. Any number of 3-D objects may lie on line w and the distance between O and the vertices O1 to On may be defined by w.
Example: A train engine defined by 3-D co-ordinates on a track, with no branches. The track is the 4th dimension, and if a station is located on the vertex of the straight line, then the distance between the train and the station is given by the 4th dimension.
5-D Space
Five dimensional space is the 3-D space defined above whose vertex lies in a 2-D plane. Therefore, in 5-D space at least two vertices exist and the distance between them is defined by the 4th and 5th dimensions.
6-D Space
Six dimensional space is the 3-D space whose vertex lies in another 3-D space. In 6-D space at least two vertices exist and the distance between the two is defined by the 4th, 5th and 6th dimensions.
7-D Space
In seven dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O1 and O2 where O1 lies along on a straight line s with vertex O. Let the following be a six dimensional space:
Then, the 7-D space is as follows:
8-D Space
In eight dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O1 and O2 where O1 lies along on a plane with vertex O, and axis s and r.
9-D Space
In nine dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O1 and O2 where O1 lies on the 3-D space with vertex O and axis s, r and q.
n-D Space
The above pattern can be nested to infinite levels, in sets of three. The sun, earth and moon can be plotted without any ambiguity in nine dimensions. Other dimensions may be added in sets of three, down to fundamental particles, or higher up to other galaxies, and their relationships may be mapped and studied with more clarity.
For example a 12-D space would be as follows:
A higher number of dimensions can always be expressed in lesser number of dimensions. For example, no matter how many levels deep, they can be plotted in a 2-D plane of the screen, or a 1-D line of bits and components.
Arrays can be used to efficiently store information about the dimensional space. The dimensions higher than 3 spatially relate to mod 3 and can be represented as a graph with parent and child nodes. Any two nodes may be singled out as frames of reference.
Last modified: Fri Jul 14 05:21:45 IST 2017