Tiger Cave and the Mahabalipuram Complex

Tiger Cave:

Tiger cave a disappointment: “I was disappointed on visiting this place, it does not have any tourist attraction, and there is no such cave here.” - Someone on tripadvisor.in

‘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.

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.

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.

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.

The turtle yali module symbolizes flying out to space by those who acquire vingyan/knowledge that stems from வின்+ஞானம் - the others stay grounded here like the gorilla and the whales, leaping only that high.

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 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.

Yet another short proof for Fermat's Last Theorem

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 that
a^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^3
Take 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^2In cases where  d^2+e^2>f^2, it corresponds to an acute triangle on a plane and it is possible that d^3 = a^3 + b^3 + c^3.
For 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^2For eg. 6^3 + 8^3 + 10^3 = 12^3. In this case, 6^2 + 8^2 < 12^2,  6^2 + 10^2 < 12^2 forming obtuse triangles and 8^2 + 10^2 > 12^2 forming an acute triangle.

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.

Pictorial proof for Fermat's Last Theorem

Pythagoras Theorem

c^2 = a^2 + b^2

Fermat's Last Theorem

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.

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:  

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

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.

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.

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.