Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
mNo edit summary
No edit summary
Line 1: Line 1:
The '''Cartan decomposition''' is a decomposition of a [[Semisimple Lie algebra|semisimple]] [[Lie group]] or [[Lie algebra]], which plays an important role in their structure theory and [[representation theory]]. It generalizes the [[polar decomposition]] or [[singular value decomposition]] of matrices. Its history can be traced to the 1880s work of [[Élie Cartan]] and [[Wilhelm Killing]]. [http://books.google.com/books?id=udj-1UuaOiIC&pg=PA46&dq=history+cartan+decomposition&hl=en&sa=X&ei=aa-wUuCDEMGmkQfNqoHABg&ved=0CDQQ6AEwAQ#v=onepage&q=history%20cartan%20decomposition&f=false]
{{Hinduism}}
{{Pi box}}
'''Baudhāyana''', (fl. c. 800 BCE)<ref name="Baudhayana">O'Connor, "Baudhayana".</ref> was the author of the Baudhayana sūtras, which cover dharma, daily ritual, Vedic sacrifices, etc. He belongs to the [[Yajurveda]] school, and is older than the other sūtra author [[Apastamba|Āpastambha]].


== Cartan involutions on Lie algebras ==
He was the author of the earliest ''[[Sulba Sutras|Sulba Sūtra]]''—appendices to the [[Vedas]] giving rules for the construction of [[altars]]—called the ''{{IAST |Baudhāyana Śulbasûtra}}''. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of [[pi]] to some degree of precision, and stating a version of what is now known as the [[Pythagorean theorem]].


Let <math>\mathfrak{g}</math> be a real [[Semisimple Lie algebra|semisimple]] [[Lie algebra]] and let <math>B(\cdot,\cdot)</math> be its [[Killing form]].  An [[Involution (mathematics)|involution]] on <math>\mathfrak{g}</math> is a Lie algebra [[automorphism]] <math>\theta</math> of <math>\mathfrak{g}</math> whose square is equal to the identity. Such an involution is called a '''Cartan involution''' on <math>\mathfrak{g}</math> if <math>B_\theta(X,Y) := -B(X,\theta Y)</math> is a [[positive definite bilinear form]].
==The sūtras of Baudhāyana==
The {{IAST|Sûtras}} of {{IAST|Baudhāyana}} are associated with the ''Taittiriya'' {{IAST|Śākhā}} (branch) of Krishna (black) ''[[Yajurveda]]''. The sutras of {{IAST|Baudhāyana}} have six sections,
# the [[Srautasutra|{{IAST|Śrautasûtra}}]], probably in 19 {{IAST|Praśnas}} (questions),  
# the {{IAST|Karmāntasûtra}} in 20 {{IAST|Adhyāyas}} (chapters),
# the {{IAST|Dvaidhasûtra}} in 4 {{IAST|Praśnas}},
# the [[Grhyasutra|Grihyasutra]] in 4 {{IAST|Praśnas}},
# the [[Dharmasutra|{{IAST|Dharmasûtra}}]] in 4 {{IAST|Praśnas}} and
# the [[Sulba Sutras|{{IAST|Śulbasûtra}}]] in 3 {{IAST|Adhyāyas}}.<ref>[http://www.sacred-texts.com/hin/sbe14/sbe1403.htm ''Sacred Books of the East'', vol.14 – Introduction to Baudhayana]</ref>


Two involutions <math>\theta_1</math> and <math>\theta_2</math> are considered equivalent if they differ only by an [[inner automorphism]].
===The Shrautasūtra===
{{main|Baudhayana Shrauta Sutra}}
His [[Śrauta|shrauta]] [[sūtra]]s related to performing [[Vedas|Vedic]] [[sacrifice]]s has followers in some [[Smartism|Smārta]] [[brāhmaṇa]]s ([[Iyers]]) and some [[Iyengar]]s and [[Gounder|Kongu]] of [[Tamil Nadu]], [[Yajurveda|Yajurvedis]] or [[Namboothiri]]s of [[Kerala]], Gurukkal Brahmins, among others.  The followers of this sūtra follow a different method and do 24 Tila-tarpaṇa, as  Lord [[Krishna]] had done tarpaṇa on the day before [[Amavasya|amāvāsyā]];  they call themselves Baudhāyana Amavasya.


Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.
===The Dharmasūtra===
The Dharmasūtra of Baudhāyana like that of [[Apastamba]] also forms a part of the larger [[Kalpasutra]]. Likewise, it is composed of ''[[praśna]]s'' which literally means ‘questions’ or books. The structure of this Dharmasūtra is not very clear because it came down in an incomplete manner. Moreover, the text has undergone alterations in the form of additions and explanations over a period of time. The ''praśnas'' consist of the [[Srautasutra]] and other ritual treatises, the Sulvasutra which deals with vedic geometry, and the [[Grhyasutra]] which deals with domestic rituals.<ref name="Patrick Olivelle 1999 p.127">Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p.127</ref>


=== Examples ===
====Authorship and Dates====
Āpastamba and Baudhāyana come from the [[Taittiriya]] branch vedic school dedicated to the study of the [[Black Yajurveda]]. [[Robert Lingat]] states that Baudhāyana was the first to compose the Kalpasūtra collection of the Taittiriya school followed by Āpastamba.<ref>Robert Lingat, The Classical Law of India, (Munshiram Manoharlal Publishers Pvt Ltd, 1993), p.20</ref> Kane assigns this Dharmasūtra an approximate date between 500 to 200 BC.<ref name="Patrick Olivelle 1999">Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p.xxxi</ref>


{{^|NOTE: Blank lines between items helped source readability, but screwed up list formatting}}
====Commentaries====
* A Cartan involution on <math>\mathfrak{sl}_n(\mathbb{R})</math> is defined by <math>\theta(X)=-X^T</math>, where <math>X^T</math> denotes the transpose matrix of <math>X</math>.
There are no commentaries on this Dharmasūtra with the exception of [[Govindasvāmin]]'s ''Vivaraṇa''. The date of the commentary is uncertain but according to Olivelle it is not very ancient. Also the commentary is inferior in comparison to that of Haradatta on Āpastamba and Gautama.<ref name="Patrick Olivelle 1999"/>
* The identity map on <math>\mathfrak{g}</math> is an involution, of course.  It is the unique Cartan involution of <math>\mathfrak{g}</math> if and only if the Killing form of <math>\mathfrak{g}</math> is negative definite.  Equivalently, <math>\mathfrak{g}</math> is the Lie algebra of a compact semisimple Lie group.
* Let <math>\mathfrak{g}</math> be the complexification of a real semisimple Lie algebra <math>\mathfrak{g}_0</math>, then complex conjugation on <math>\mathfrak{g}</math> is an involution on <math>\mathfrak{g}</math>.  This is the Cartan involution on <math>\mathfrak{g}</math> if and only if <math>\mathfrak{g}_0</math> is the Lie algebra of a compact Lie group.
* The following maps are involutions of the Lie algebra <math>\mathfrak{su}(n)</math> of the special unitary group [[SU(n)]]:
** the identity involution <math>\theta_0(X) = X</math>, which is the unique Cartan involution in this case;
** <math>\theta_1 (X) = - X^T</math> which on <math>\mathfrak{su}(n)</math> is also the complex conjugation;
** if <math>n = p+q</math> is odd, <math>\theta_2 (X) = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix} X \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}</math>. These are all equivalent, but not equivalent to the identity involution (because the matrix <math>\begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}</math> does not belong to <math>\mathfrak{su}(n)</math>.)
** if <math>n = 2m</math> is even, we also have <math>\theta_3 (X) = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix} X^T \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}.</math>


== Cartan pairs ==
====Organization and Contents====
This Dharmasūtra is divided into four books. Olivelle states that Book One and the first sixteen chapters of Book Two are the ‘Proto-Baudhayana’<ref name="Patrick Olivelle 1999 p.127" /> even though this section has undergone alteration. Scholars like Bühler and Kane agree that the last two books of the Dharmasūtra are later additions. Chapter 17 and 18 in Book Two lays emphasis on various types of ascetics and acetic practices.<ref name="Patrick Olivelle 1999 p.127" />


Let <math>\theta</math> be an involution on a Lie algebra <math>\mathfrak{g}</math>. Since <math>\theta^2=1</math>, the linear map <math>\theta</math> has the two eigenvalues <math>\pm1</math>. Let <math>\mathfrak{k}</math> and <math>\mathfrak{p}</math> be the corresponding eigenspaces, then <math>\mathfrak{g} = \mathfrak{k}+\mathfrak{p}</math>. Since <math>\theta</math> is a Lie algebra automorphism, eigenvalues are multiplicative. It follows that
The first book is primarily devoted to the student and deals in topics related to studentship. It also refers to social classes, the role of the king, marriage, and suspension of Vedic recitation. Book two refers to penances, inheritance, women, householder, orders of life, ancestral offerings. Book three refers to holy householders, forest hermit and penances. Book four primarily refers to the yogic practices and penances along with offenses regarding marriage.<ref>Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p.128-131</ref>
: <math>[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k}</math>, <math>[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}</math>, and <math>[\mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}</math>.
Thus <math>\mathfrak{k}</math> is a Lie subalgebra, while any subalgebra of <math>\mathfrak{p}</math> is commutative.


Conversely, a decomposition <math>\mathfrak{g} = \mathfrak{k}+\mathfrak{p}</math> with these extra properties determines an involution <math>\theta</math> on <math>\mathfrak{g}</math> that is <math>+1</math> on <math>\mathfrak{k}</math> and <math>-1</math> on <math>\mathfrak{p}</math>.
==The mathematics in Sulbasūtra==


Such a pair <math>(\mathfrak{k}, \mathfrak{p})</math> is also called a '''Cartan pair''' of <math>\mathfrak{g}</math>.
===Pythagorean theorem===
The most notable of the rules (the Sulbasūtra-s do not contain any proofs of the rules which they describe, since they are sūtra-s, formulae, concise) in the ''Baudhāyana Sulba Sūtra'' says:
<blockquote>
''dīrghasyākṣaṇayā rajjuḥ pārśvamānī, tiryaḍam mānī,''<br>
''cha yatpṛthagbhūte kurutastadubhayāṅ karoti.''<br>
:A rope stretched along the length of the [[diagonal]] produces an [[area]] which the vertical and horizontal sides make together.''{{Citation needed|date=November 2013}}
</blockquote>


The decomposition <math>\mathfrak{g} = \mathfrak{k}+\mathfrak{p}</math> associated to a Cartan involution is called a '''Cartan decomposition''' of <math>\mathfrak{g}</math>.  The special feature of a Cartan decomposition is that the Killing form is negative definite on <math>\mathfrak{k}</math> and positive definite on <math>\mathfrak{p}</math>.  Furthermore, <math>\mathfrak{k}</math> and <math>\mathfrak{p}</math> are orthogonal complements of each other with respect to the Killing form on <math>\mathfrak{g}</math>.
This appears to be referring to a rectangle, although some interpretations consider this to refer to a [[Square (geometry)|square]].  In either case, it states that the square of the [[hypotenuse]] equals the sum of the squares of the sides.  If restricted to right-angled isosceles triangles, however, it would constitute a less general claim, but the text seems to be quite open to unequal sides.


== Cartan decomposition on the Lie group level ==
If this refers to a rectangle, it is the earliest recorded statement of the [[Pythagorean theorem]].


Let <math>G</math> be a [[Semisimple Lie group|semisimple]] [[Lie group]] and <math>\mathfrak{g}</math> its [[Lie algebra]].  Let <math>\theta</math> be a Cartan involution on <math>\mathfrak{g}</math> and let <math>(\mathfrak{k},\mathfrak{p})</math> be the resulting Cartan pair.  Let <math>K</math> be the [[analytic subgroup]] of <math>G</math> with Lie algebra <math>\mathfrak{k}</math>.  Then:
Baudhāyana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles [[right triangle]]:
* There is a Lie group automorphism <math>\Theta</math> with differential <math>\theta</math> that satisfies <math>\Theta^2=1</math>.
* The subgroup of elements fixed by <math>\Theta</math> is <math>K</math>; in particular, <math>K</math> is a closed subgroup.
* The mapping <math>K\times\mathfrak{p} \rightarrow G</math> given by <math>(k,X) \mapsto k\cdot \mathrm{exp}(X)</math> is a diffeomorphism.
* The subgroup <math>K</math> contains the center <math>Z</math> of <math>G</math>, and <math>K</math> is compact modulo center, that is, <math>K/Z</math> is compact.
* The subgroup <math>K</math> is the maximal subgroup of <math>G</math> that contains the center and is compact modulo center.


The automorphism <math>\Theta</math> is also called '''global Cartan involution''', and the diffeomorphism <math>K\times\mathfrak{p} \rightarrow G</math> is called '''global Cartan decomposition'''.
:''The cord which is stretched across a square produces an area double the size of the original square.''


For the general linear group, we get <math> X \mapsto (X^{-1})^T </math> as the Cartan involution.
===Circling the square===
Another problem tackled by Baudhāyana is that of finding a circle whose area is the same as that of a square (the reverse of [[squaring the circle]]). His sūtra i.58 gives this construction:


A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras <math>\mathfrak{a}</math> in <math>\mathfrak{p}</math> are unique up to conjugation by ''K''. Moreover
:''Draw half its diagonal about the centre towards the East-West line; then describe a circle together with a third part of that which lies outside the square. ''


:<math>\displaystyle{\mathfrak{p}= \bigcup_{k\in K} \mathrm{Ad}\, k \cdot \mathfrak{a}.}</math>
Explanation:
*Draw the half-diagonal of the square, which  is larger than the half-side by <math>x = {a \over 2}\sqrt{2}- {a \over 2}</math>.
*Then draw a circle with radius <math>{a \over 2} + {x \over 3}</math>, or <math>{a \over 2} + {a \over 6}(\sqrt{2}-1)</math>, which equals <math>{a \over 6}(2 + \sqrt{2})</math>.
* Now <math>(2+\sqrt{2})^2 \approx 11.66 \approx {36.6\over \pi}</math>, so the area <math>{\pi}r^2 \approx \pi \times {a^2 \over 6^2} \times {36.6\over \pi} \approx a^2</math>.


In the compact and noncompact case this Lie algebraic result implies the decomposition
===Square root of 2===
Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6)
gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:


:<math>\displaystyle{G=KAK,}</math>
:''samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet <br> tac caturthenātmacatustriṃśonena saviśeṣaḥ''


where ''A'' = exp <math>\mathfrak{a}</math>. Geometrically the image of the subgroup ''A'' in ''G'' / ''K'' ia a [[totally geodesic]] submanifold.
: The diagonal [lit. "doubler"] of a square. The measure is to be increased by a third and by a fourth decreased by the 34th. That is its diagonal approximately.{{citation needed|date=December 2012}}


== Relation to polar decomposition ==
That is,


Consider <math>\mathfrak{gl}_n(\mathbb{R})</math> with the Cartan involution <math>\theta(X)=-X^T</math>.  Then <math>\mathfrak{k}=\mathfrak{so}_n(\mathbb{R})</math> is the real Lie algebra of skew-symmetric matrices, so that <math>K=\mathrm{SO}(n)</math>, while <math>\mathfrak{p}</math> is the subspace of symmetric matrices.  Thus the exponential map is a diffeomorphism from <math>\mathfrak{p}</math> onto the space of positive definite matrices.  Up to this exponential map, the global Cartan decomposition is the [[polar decomposition]] of a matrix.  Notice that the polar decomposition of an invertible matrix is unique.
:<math>\sqrt{2} \approx  1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot4 \cdot 34} = \frac{577}{408} \approx 1.414216,</math>


== See also ==
which is correct to five decimals.<ref name="Baudhayana"/>


* [[Lie group decompositions]]
Other theorems include: diagonals of rectangle bisect each other,
diagonals of rhombus bisect at right angles, area of a square formed
by joining the middle points of a square is half of original, the
midpoints of a rectangle joined forms a rhombus whose area is half the
rectangle, etc.


== References ==
Note the emphasis on rectangles and squares; this arises from the need
* {{citation|first=Sigurdur|last= Helgason|title=Differential geometry, Lie groups, and symmetric spaces|year=1978|publisher=Academic Press|isbn= 0-8218-2848-7}}
to specify ''yajña bhūmikā''s—i.e. the altar on which a rituals were
*[[A. W. Knapp]], ''Lie groups beyond an introduction'', ISBN 0-8176-4259-5, Birkhäuser.
conducted, including fire offerings (yajña).


[[Category:Lie groups]]
[[Āpastamba]] (c. 600 BC) and [[Kātyāyana]] (c. 200 BC), authors of other sulba sūtras, extend some of Baudhāyana's ideas. Āpastamba provides a more general proof{{Citation needed|date=February 2007}} of the Pythagorean theorem.
[[Category:Lie algebras]]
 
==Notes==
<div class="references-small">
<references/>
 
==See also==
*[[Indian mathematics]]
*[[Indian mathematicians]]
*[[Sulba Sutras]]
 
==References==
* George Gheverghese Joseph. ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd Edition. [[Penguin Books]], 2000. ISBN 0-14-027778-1.
* Vincent J. Katz. ''A History of Mathematics: An Introduction'', 2nd Edition. [[Addison-Wesley]], 1998. ISBN 0-321-01618-1
* [[S. Balachandra Rao]], ''Indian Mathematics and Astronomy: Some Landmarks''. Jnana Deep Publications, Bangalore, 1998. ISBN 81-900962-0-6
* {{MacTutor Biography|id=Baudhayana}} [[St Andrews University]], 2000.
* {{MacTutor Biography|id=Indian_sulbasutras|title=The Indian Sulbasutras|class=HistTopics}} St Andrews University, 2000.
* Ian G. Pearce. [http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch4_2.html ''Sulba Sutras''] at the [[MacTutor archive]]. St Andrews University, 2002.
</div>
*B.B.DUTTA."THE SCIENCE OF THE SULBA".
 
{{Indian mathematics}}
 
{{DEFAULTSORT:Baudhayana}}
[[Category:Ancient Indian mathematicians]]
[[Category:Pi]]
[[Category:Indian_mathematics]]

Revision as of 07:47, 13 August 2014

Systems Analyst Carmouche from Manitou, has interests for instance metal detecting, property developers ec in singapore singapore and sky diving. During the recent few months has made a trip to places for example Chongoni Rock-Art Area. Template:Pi box Baudhāyana, (fl. c. 800 BCE)[1] was the author of the Baudhayana sūtras, which cover dharma, daily ritual, Vedic sacrifices, etc. He belongs to the Yajurveda school, and is older than the other sūtra author Āpastambha.

He was the author of the earliest Sulba Sūtra—appendices to the Vedas giving rules for the construction of altars—called the Template:IAST. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of pi to some degree of precision, and stating a version of what is now known as the Pythagorean theorem.

The sūtras of Baudhāyana

The Template:IAST of Template:IAST are associated with the Taittiriya Template:IAST (branch) of Krishna (black) Yajurveda. The sutras of Template:IAST have six sections,

  1. the [[Srautasutra|Template:IAST]], probably in 19 Template:IAST (questions),
  2. the Template:IAST in 20 Template:IAST (chapters),
  3. the Template:IAST in 4 Template:IAST,
  4. the Grihyasutra in 4 Template:IAST,
  5. the [[Dharmasutra|Template:IAST]] in 4 Template:IAST and
  6. the [[Sulba Sutras|Template:IAST]] in 3 Template:IAST.[2]

The Shrautasūtra

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. His shrauta sūtras related to performing Vedic sacrifices has followers in some Smārta brāhmaṇas (Iyers) and some Iyengars and Kongu of Tamil Nadu, Yajurvedis or Namboothiris of Kerala, Gurukkal Brahmins, among others. The followers of this sūtra follow a different method and do 24 Tila-tarpaṇa, as Lord Krishna had done tarpaṇa on the day before amāvāsyā; they call themselves Baudhāyana Amavasya.

The Dharmasūtra

The Dharmasūtra of Baudhāyana like that of Apastamba also forms a part of the larger Kalpasutra. Likewise, it is composed of praśnas which literally means ‘questions’ or books. The structure of this Dharmasūtra is not very clear because it came down in an incomplete manner. Moreover, the text has undergone alterations in the form of additions and explanations over a period of time. The praśnas consist of the Srautasutra and other ritual treatises, the Sulvasutra which deals with vedic geometry, and the Grhyasutra which deals with domestic rituals.[3]

Authorship and Dates

Āpastamba and Baudhāyana come from the Taittiriya branch vedic school dedicated to the study of the Black Yajurveda. Robert Lingat states that Baudhāyana was the first to compose the Kalpasūtra collection of the Taittiriya school followed by Āpastamba.[4] Kane assigns this Dharmasūtra an approximate date between 500 to 200 BC.[5]

Commentaries

There are no commentaries on this Dharmasūtra with the exception of Govindasvāmin's Vivaraṇa. The date of the commentary is uncertain but according to Olivelle it is not very ancient. Also the commentary is inferior in comparison to that of Haradatta on Āpastamba and Gautama.[5]

Organization and Contents

This Dharmasūtra is divided into four books. Olivelle states that Book One and the first sixteen chapters of Book Two are the ‘Proto-Baudhayana’[3] even though this section has undergone alteration. Scholars like Bühler and Kane agree that the last two books of the Dharmasūtra are later additions. Chapter 17 and 18 in Book Two lays emphasis on various types of ascetics and acetic practices.[3]

The first book is primarily devoted to the student and deals in topics related to studentship. It also refers to social classes, the role of the king, marriage, and suspension of Vedic recitation. Book two refers to penances, inheritance, women, householder, orders of life, ancestral offerings. Book three refers to holy householders, forest hermit and penances. Book four primarily refers to the yogic practices and penances along with offenses regarding marriage.[6]

The mathematics in Sulbasūtra

Pythagorean theorem

The most notable of the rules (the Sulbasūtra-s do not contain any proofs of the rules which they describe, since they are sūtra-s, formulae, concise) in the Baudhāyana Sulba Sūtra says:

dīrghasyākṣaṇayā rajjuḥ pārśvamānī, tiryaḍam mānī,
cha yatpṛthagbhūte kurutastadubhayāṅ karoti.

A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

This appears to be referring to a rectangle, although some interpretations consider this to refer to a square. In either case, it states that the square of the hypotenuse equals the sum of the squares of the sides. If restricted to right-angled isosceles triangles, however, it would constitute a less general claim, but the text seems to be quite open to unequal sides.

If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem.

Baudhāyana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles right triangle:

The cord which is stretched across a square produces an area double the size of the original square.

Circling the square

Another problem tackled by Baudhāyana is that of finding a circle whose area is the same as that of a square (the reverse of squaring the circle). His sūtra i.58 gives this construction:

Draw half its diagonal about the centre towards the East-West line; then describe a circle together with a third part of that which lies outside the square.

Explanation:

Square root of 2

Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6) gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:

samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet
tac caturthenātmacatustriṃśonena saviśeṣaḥ
The diagonal [lit. "doubler"] of a square. The measure is to be increased by a third and by a fourth decreased by the 34th. That is its diagonal approximately.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

That is,

2≈1+13+13⋅4−13⋅4⋅34=577408≈1.414216,

which is correct to five decimals.[1]

Other theorems include: diagonals of rectangle bisect each other, diagonals of rhombus bisect at right angles, area of a square formed by joining the middle points of a square is half of original, the midpoints of a rectangle joined forms a rhombus whose area is half the rectangle, etc.

Note the emphasis on rectangles and squares; this arises from the need to specify yajña bhūmikās—i.e. the altar on which a rituals were conducted, including fire offerings (yajña).

Āpastamba (c. 600 BC) and Kātyāyana (c. 200 BC), authors of other sulba sūtras, extend some of Baudhāyana's ideas. Āpastamba provides a more general proofPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. of the Pythagorean theorem.

Notes

  1. ↑ 1.0 1.1 O'Connor, "Baudhayana".
  2. ↑ Sacred Books of the East, vol.14 – Introduction to Baudhayana
  3. ↑ 3.0 3.1 3.2 Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p.127
  4. ↑ Robert Lingat, The Classical Law of India, (Munshiram Manoharlal Publishers Pvt Ltd, 1993), p.20
  5. ↑ 5.0 5.1 Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p.xxxi
  6. ↑ Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p.128-131

See also

References

  • B.B.DUTTA."THE SCIENCE OF THE SULBA".

Template:Indian mathematics

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=49464"