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LATEST DIALOGUES Art & Spirit in Mathematics: The Lessons of Japanese Temple Geometry (part II)

Photo by Florian Holub

Photo by Florian Holub

Three Temple Geometry Problems Presented on Beautiful Trees
I am an occasional artist, and have been working for several years on a series of slides now called Japanese Temple Geometry on Beautiful Trees. Geometric diagrams taken from the sangaku are re-created with the GeoGebra program, and placed on photographs of trees (mostly mine) with GIMP (Gnu Image Manipulation Program). This is, of course, an homage to the way that sangaku diagrams were painted on wood. The three problems whose mathematics we will consider are introduced by these slides, in the hope of encouraging a response that transcends the intellectual, perhaps resonating with the aesthetic and spiritual pleasure that the Temple Geometers took in them.

I encourage you to spend some time contemplating the slides before going on to an explanation of the problem and a description of its solution. And also, if you choose to delve into the mathematical side, to return to the slides with the aim of keeping your aesthetic and spiritual appreciation of the problem from being overwhelmed by the intellectual side of the math.

To give a more tangible sense of JTG, we will explore three problems, all from Fukagawa & Rothman’s Chapter 4, Easier Temple Geometry Problems. They are problems with relatively low mathematical complexity that I find beautiful, both visually and conceptually.

They were also chosen to illustrate an interesting tension in the ethos of Temple Geometry. On one hand, there is an intense love of the geometric image, painted in color and hung in a sacred space as an object of contemplation. But on the other hand, there seems to be greater ease with arithmetic and algebraic computations than with creative geometric thinking, and the chosen problems illustrate this.

Problem A. A Surprising Symmetry of Circles (F&R pp 100-101).
The first slide, a shimmering set of sixteen circles, two sets of eight shown twice, gives a sense of what is fixed and what moves in this dynamic problem. Both slides give multiple snapshots of the process, though in very different ways, with the first superimposing two views, and the second giving two separate views, adding a slice of a third. The problem asks the question: Are three circles all the same size? The answer is yes, and it is a surprising symmetry because the construction of the center one is much simpler than that of the outer two. The solution we give was found by the Temple Geometers, and we have not substantially improved on the presentation of Fukagawa & Rothman. It is quite elementary, but it requires drawing some additional lines and using the Pythagorean theorem twice, and then doing some algebra. I have not succeeded in finding a solution that gives a satisfying geometric understanding of the symmetry.

B. The Broken Symmetry of the Pentagram, Problem 33, pp 111-112. This problem is introduced with a single slide of a tree in Regent’s Park 4-isosceles-triangleswhose somewhat grotesque beauty seems to invoke the broken symmetry of the pentagram. In contrast to Problem A, the central circle in this case is slightly larger than the circles on the periphery. The problem asks for the size of the break, the ratio of outer to central circles. As we’ll see, the pentagram is thoroughly structured by the Golden Ratio, I wanted to see how θ was used in finding the symmetry-breaking ratio. But the JTG solution given by Fukagawa & Rothman never mentions anything golden, and instead uses some esoteric trigonometry. The solution we give follows F & R, except that the trigonometry is replaced by use of golden triangles. As seen here, there are two golden triangles, isosceles triangles in which the ratio of the longer side to shorter side is always θ. The one on the left is acute, and the one on the right is obtuse.

C. The Elegant Economy of the Susaka Triangle, Problem 47, pp 120-121. Two slides with very different moods introduce this dynamic problem, each superimposing three moments in its growth. One is clear, sharp, reaching dynamically upward. The other, which also contains a diagram of the solution, is murky and mysterious. I love the simplicity of the problem, which contains one right triangle, one circular arc, and one square, and asks a question that has a beautiful answer. The solution of the Temple Geometers was not particularly beautiful. It involved their version of differential calculus. An expression is found for the side of the square in terms of the height of the triangle, and the derivative is set to zero to find the maximum. The answer is agreeably simple, but there is no geometric sense of why it should come out so. F & R present both the JTG proof and a modern proof, which is a model of geometric thinking. Once you really see the diagram, the answer becomes crystal clear. We give an expanded version of the modern proof.

A. A Surprising Symmetry of Circles, Problem 16, pp 100-101.

5-circles

Photo by Michele Laporte

The left slide contains two snapshots of the dynamic process, which are superimposed. The right slide contains two full snapshots and a slice of a third, but they are distributed. Can you see what the problem might be? Can you separate the two snapshots? Can you see what moves and what stands still?

Problem A
Let’s look at the elements of this problem. There are eight circles in all, but everything is determined by the four red circles of radius r, arranged with their centers in a rectangle. Horizontally, each pair meet in a point, so the rectangle has width 2r, but the vertical dimension, which we call 2t, 6-problem-A-circlescan change. We draw a large red circle around the red circles and a small green circle in the center. Finally we draw two small blue circles between the red and the large green, so that everything touches but doesn’t cross. By symmetry, the left and right blue circles have the same radius q, and we denote the radius of the central circle by p.

The problem asks the question: What is the relationship between p and q? They are very close, but are they actually equal? One might suspect that they are different, since the central green circle only requires the four red circles of radius r, but the two blue circles require that the large red circle be drawn first.

Solution A
The answer is yes, p does equal q. First we add a few lines to the eight circles, forming a triangle with vertices at the centers of three circles: center, upper left, and left, and then divide it into two right triangles by dropping a perpendicular. The hypotenuse of one triangle is r+q and of the other r+p. The proof uses the Pythagorean Theorem twice, once on each of the small right triangles, followed by a bit of algebraic manipulation. Let R denote the radius of the large circle.

Applying the Pythagorean Theorem to the triangle on the right gives:

7-problem-A-circles

This demonstration fails to satisfy me. While I can follow the proof and justify every step, it doesn’t answer: Why is the green circle the same size as the blue ones? I have failed to find a geometric argument that gives me a deeper understanding of the geometric construction, which this computation does not do. Nevertheless, I enjoy the problem, and invite you to revisit the two slides now that you have seen the math behind it.

Let me mention one issue that I glossed over above. Once the four red circles are in position, it is easy to locate the center point to draw the small green and large red circles. But it is not obvious how locate the centers of the blue circles, which are tangent to two red circles of radius r, and to the surrounding circle. The problem of constructing a circle tangent to three given circles was not solved by Euclid, but by his great successor Apollonius of Perga, who lived from 262 to 190 BCE. A good reference for his methods is: www.cut-the-knot.org/pythagoras/Apollonius.shtml
Fortunately, in this case we don’t need Appolonius’ method. Once we know that p=q we use the radius of the small green circle to locate the centers of the blue circles.

B. The Broken Symmetry of the Pentagram 

8-pentagramThe obvious connection between the pentagram and the tree is rotundity, with the tree being an organic explosion of the pristine perfection of the circle, which itself may seem pressed outward by the pentagram and the five circles in its arms. I love the tree for its unruly growth, for breaking the symmetry of a gradually tapering trunk. And the pentagram itself carries a broken symmetry, which is at the heart of this problem. Do you see it?

Problem B
You may have noticed that the green circle in the center appears slightly larger than the five blue circles, which are the same size because of symmetry. That symmetry is broken by the green circle, which is indeed slightly larger. The problem posed on the sangaku is to find out precisely how much larger the central green circle is than the blue circles to find an expression for r, the radius of the green circle, in terms of t, the radius of the blues.

Note that the symmetry that is broken here is rather similar to that which was established in Problem A, as both involve the relationship of a central circle to peripheral ones. When I first considered this problem I was a bit offended. How dare the circles come so close to being equal and yet not actually be the same? Unfair! Then I came to appreciate the broken symmetry, which seems central to placing the figure on this special tree, and is intimately connected with the Golden Ratio.

9-problem-B-circlesFirst, let’s look at the construction of this figure. GeoGebra draws the pentagon automatically, and it is not hard to do it by drawing circles and straight lines, i.e. using ruler and compass. Then the pentagram is an easy construction, as are the small and large green circles. But how do we locate the centers of the five blue circles? That question is answered by the dashed lines at the bottom of the diagram. We extend two of the edges of the pentagram and draw a tangent to the bottom of the circle to form the triangle ΔPXY. Then we can locate the center C of the bottom blue circle by bisecting the angle X and intersecting it with the central line of the pentagram.

Solution B
The answer is that the radius r of the central circle is

10-formula

 

 

This means that the radius of the central circle is about 12% larger than the radii of the blue circles, and the area of the central circle is exactly 25% larger than that of the blues.

If you’re curious about how this answer is obtained, and about the role played by the Golden Ratio and golden triangles, read on. Unlike the JTG proof of the Temple Geometers given by Fukagawa and Rothman, our proof will not use any fancy trigonometry, or any trig at all. The trig is replaced by the golden triangles. First we’ll take a look at how the pentagram is structured by θ. Forget for a moment that you know anything about the Golden Ratio. Everything you need to know about θ can be read off of the pentagram when it’s inscribed in a pentagon, as below:

11-pentagram-diagramTake a good look at this diagram of the pentagram in a pentagon. The angles can all be deduced from the symmetry of the figure.

For simplicity, we set each edge of the inner pentagon to length 1, and now for a moment imagine that θ is not the Golden Ratio, but just the symbol that we happen to use for edges like EF, of which there are 10 (not all labeled). Now notice the parallelogram ABGE, whose top edge has length θ+1, so that the bottom, which is a side of the pentagon, has the same length.

Now notice the triangles ΔFGD and ΔABD. Since they are similar, and the small one is a golden triangle, so is ΔABD, whose long sides are 2θ+1 with short side θ+1. Since

12-formula

 

 

which yields two magic formulas that define the Golden Ratio:

13-formula

 

 

This justifies our use of the symbol θ, and shows that each side of the large pentagon has length θ2=θ+1. We can also find that

14-formula

 

 

which means that the diagonals of the pentagram have length θ3, so every line segment in the figure has length in the set {1, θ, θ2, θ3}. Furthermore, we can now see that every triangle in the pentagram is golden. We now know the angles of the golden triangles.

15-triangle1

 

 

 

 

 

 

Because we will encounter them below, we also look at the right triangles obtained by dividing the golden triangles down the middle.

16-triangle2

 

 

 

 

 

 

 

Turning back to the main problem, we start by drawing the diagonal HD, the chord HC and the radius OC, and then drop perpendiculars from P to OC and O to HC. Because it’s one tenth of a full circle, ∠DOC = 36o degrees, and ∠DHC = 18o degrees, half of a pentagram vertex. Notice the two shaded right triangles ΔOWP, which has an angle of 36o degrees and ΔOZH, which has an angle of 18o degrees. Each triangle is half of a golden triangle, so the ratios of corresponding sides will be the same. We let R denote the radius of the large red circle. Then, comparing triangle ΔOZH with the acute half golden triangle gives:

17-problem-b-circles

18-formula

 

 

Comparing ΔOWP with the obtuse half golden triangle gives

19-formula

 

 

Using (1) and (2) to eliminate R

20-formula

 

 

Multiplying by θ and using θ2 = θ + 1 gives

21-formula

 

 

which simplifies to

22-formula

 

 

or

23-formula

 

 

Now use

24-formula

 

 

to simplify:

25-formula

 

 

So, if you like your answer in terms of the Golden Ratio:

27-formula

 

 

Since

26-formula

 

 

we have

28-formula

 

 

which puts the difference between r and t in another form:

29-formula

 

 

 

Congratulations if you hung in to the end of the proof.

For those comfortable with a bit of trigonometry, notice that we have shown these formulas for the angles of golden triangles:

30-formula

 

 

 

 

<< Art & Spirit in Mathematics: The Lessons of Japanese Temple Geometry (part I)

Art & Spirit in Mathematics: The Lessons of Japanese Temple Geometry (part III) >>

 

 

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Newcomb Greenleaf was trained in mathematics, with a doctorate from Princeton, but now works in ethnomathematics, the study of math and culture, whose nonduality he celebrates. He is currently on the undergraduate faculty of Goddard College, and his previous academic position was Professor of Computer Science at Columbia. He is working on a book, "The Revolution that Failed (and Why it Matters)" about the efforts of Errett Bishop to reform mathematics.
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One Response to “Art & Spirit in Mathematics: The Lessons of Japanese Temple Geometry (part II)”

  1. November 04, 2015 at 8:22 am, dhananjay shantaram janorkar said:

    Circumference of Circle / Diameter = Goba, 6283185306 /
    2000000000 = 3.141592653 Constant of Goba means Pi is RATIONAL NUMBER.

    Hon. Chancellor, Vice-Chancellor, President, Dean, Chairman
    Board of Studies, Principal, Head of Department, Mathematics, Physics,
    Philosophy, Science and Astronomy of schools, colleges, educational institutes,
    educational boards, mathematical institutes, scientific institutes,
    universities, and the universities listed in the Q S (Quacquarelli Symonds)
    scholars and scientists in INDIA and WORLD,

    Reference:

    Editorial of International Journal of Shantaram Janorkar
    Foundation of Mathematics, Science and Spiritual. – Edition – 1, 15 September,
    2015, Volume – 1, Issue – 1, ISO 9001:2008, ISSN (P): 2454-5236, ISSN (O):
    2454-633X, Title Code: MAHBIL06980/13/1/2014-TC / For Books – ISO 9001:2008,
    ISBN: 978-81-930845-0-2, (English and Marathi). (FREE DOWNLOAD from
    http://www.sbjanorkar.com)

    (i) (Circumference of a Circle / Diameter = Goba, 6283185306 /
    2000000000 = 3.141592653 Constant of Goba). The Self – Proving
    Theorem of Goba and its Explanation on the Basis of a Formula.

    (ii) E=Mm² Which means Energy = Mass x (Speed of Mass)², Speed of Light =
    22,32,00,00,000 Mile/Second. (Twenty two Hundred and Thirty two Cores Mile/per Second).

    (iii) 504,866,505,707,712,000 (Mile)³/Second The Extent of solar system.

    (iv) The Theorem of the Formula for the
    Explanation for the Creation of the Cosmos in the Large Hadron Collider
    Machine.

    (v) Point – The Theorem
    of Existence of Point and its Aspect.

    (vi) The Distance Between
    Shining Lightening as well as Thundering Cloud and The Earth.

    (vii) To do Research on the Research Done by Shantaram Bapurao
    Janorkar we Expect to Operation from World Educational Institute, Universities,
    Scholars, Scientists.

    Note:
    Hon. Sir, Please share this all of us.

    Link for Free Online International Journal ISO 9001:2008, ISSN
    (Print & Online):- http://www.sbjanorkar.com/Gernoul.html

    Link for Free Online Books ISO 9001:2008, ISBN: –

    http://www.sbjanorkar.com/booksd.html

    Respected Sir /
    Madam,

    This international journal has been launched in memory of my
    father and researcher late Mr. Shantaram Bapurao Janorkar by “Shantaram
    Janorakar Foundation of Mathematics” to scientifically establish the research carried out by him on
    the various topics of Mathematics and I have put my best efforts in this work
    disregarding my family and my children’s future. This is the biggest sacrifice
    made by me and this sacrifice has been done by me for my country. This
    sacrifice has been done for the students all over the whole world to enable
    them to achieve the true and real knowledge. I have been doing my research work
    at different levels for the last 18 years. To really establish this research,
    on 11 December, 2011, I received valuable guidance from Hon.Prof.Dr.T.M.Karade
    (D.Sc., D.Sc., Ex. Vice-Chancellor, Indiana University, Raipur, Ex. Dean,
    Faculty of Science, Nagpur University, Nagpur) and to scientifically establish this
    research I am also receiving valuable guidance from Hon.Prof.Dr.Shriram
    B.Patil (Ex. Head, Department of Mathematics, Ex. Chairman, Board of Studies in
    Mathematics, Ex. Dean, Faculty of Science, Mumbai University, Mumbai) Through this
    guidance, I found many ways about how I should scientifically establish all
    this research and by this Institute I have launched the journal “International
    Journal of Shantaram Janorkar Foundation of Mathematics, Science and
    Spiritual.” Through this medium I have been preparing various papers on
    this research in English and Marathi languages and publishing them serially in
    this journal. I have prepared the papers on this research and I have tried to
    convert them into scientific and mathematical language without leaving the basic
    core of the research topics which were invented by my father and researcher
    late Mr. Shantaram Bapurao Janorkar and keeping intact the meaning of his
    thoughts through scientific approach, I have compiled the theorems on various
    subjects and prepared research papers on these theorems. I have prepared these
    research papers by using various systematic methods and I have tried to explain
    everything and made use of methods which are easy and quick to understand for
    you. In case you do further studies on this research and prepare papers on the
    same we shall certainly publish them in this journal free of charge and you
    will be honored with a citation and award. I desire to publish this journal in
    all regional, national and international languages.

    This journal is in ‘Print/CD-ROM/Online’ formats. I have
    been trying to deliver this journal to the scholars and scientists of 511
    universities of India and to the scholars and scientists of 10,877 universities
    all over the world. I sincerely request to honorable scholars and scientists to
    continue further my present work. There is such a great extent of knowledge in
    this research work that the yet to be in completed research will be completed
    with this research and logic and the world will get to know the real and true
    knowledge and you can put new theorems before the world created through this
    research. To
    help this real and true knowledge put forward before the world is the very
    primary objective of my efforts. It is said that time and tide waits for none; ‘Death’ is the
    eternal truth for all living beings on earth. Hence, it is utmost essential to
    put forth my research paper in front of the world. I feel, after my death,
    there will be nobody to put forth or present this research paper in front of
    the world.

    Om Purnamadah: Purnamidam Purnat Purnamudachyate I Purnasya
    Purnamadaya Purna Mevavashisyate II

    In Geometry, the
    symbol of measurement accepted (World Official) by the world community of
    scientist is degree and this same degree is the root, evidence, origin and base
    of the entire research done by my father late Mr. Shantaram Bapurao Janorkar.
    The base of this entire research is measure of circle of 36 Degree.
    By transforming this research which is originally in Marathi language, into
    scientific and mathematical language I am publishing serially all the papers on
    this research in an international journals. To begin with, I have first
    prepared the papers on the theorems:

    ® The Self – Proving Theorem of Goba and its Explanation on the
    Basis of a Formula (Circumference of a Circle / Diameter = Goba, 6283185306 /
    2000000000 = 3.141592653 Constant of Goba). (Goba Cha Swayamshidha Sidhanta Wa Sutracha Aadharacha Spastikaran
    (Wartul Parigh / Wyas = Goba, 6283185306 / 2000000000 = 3.141592653 Goba Cha
    Sthirank), (In Marathi).

    ® Point – The Theorem of Existence of Point and its Aspect. (Bindu – Binducha
    Asthitwachi Shidhatha Wa Swarup, (In Marathi).

    ® E = Mm² Which means Energy = Mass x (Speed of Mass)², Speed of Light = 22,32,00,00,000 Mile/per Second. (Twenty two
    Hundred and Thirty two Cores Mile/per Second). (E=Mm² Mhanajhach Shakati = Wastuman x
    Wastumanacha Wegacha Varga, Prakashacha Wega = 22,32,00,00,000 Mail / Prati Second (Bavis Abja Batish Koti Mail / Prati Second, (In Marathi).

    ® The Distance Between Shining Lightening as well as Thundering Cloud and
    The Earth. (Chakaknari
    Wig Aani Gadgadnara Megha yanche Pruthavi Pasun Aantar, (In Marathi),

    ® The Theorem of the Formula for the Explanation for
    the Creation of the Cosmos in the Large Hadron Collider Machine.

    ® The Theorem of the Extent of the Solar systems. (Surya malecha wyapacha shidhanta, (In
    Marathi).

    ® A Personality called Shantaram Bapurao Janorkar. (Shantaram Bapurao
    Janorkar Yak Wyaktimahatava, (In Marathi),

    ® The Establishment of Shantaram Janorkar Foundation of
    Mathematics to Create a New History in Mathematics, Science, Philosophy & Spiritual. (Ganit, Vidhyan, Thatvadhyan Vaa Adhyatma Madhye
    Yek Navin Etihas Ghadavinya Karita Shantaram Janorkar Foundation
    of Mathematics Chi Sthapana, (In Marathi),

    ® To do Research on the Research Done by Shantaram Bapurao
    Janorkar we Expect to Operation from World Educational Institute, Universities,
    Scholars, Scientists. (Shantaram Bapurao Janorkar Yancha Sanshodhanavar Sanshodhan
    Honyakarita Vishavatil Shaishanik Savunstha, Vidhyapithe, Scholars,
    Shastradhyan Kadun Sahakaryachi Aaphekasha, (In Marathi), in mathematical language and in the transforming
    the research papers on “Goba” in mathematical language by making some
    modifications, Hon.Prof.D.T.Solanke has helped me while in the translation work of this research
    work from Marathi to English, I received cooperation from Hon.Mr.Vikram K.
    More
    and Hon.Prof.Dr.T.M.Karade provided me the
    guidance on how should the first
    paragraph of the research paper on “Goba” be read. The paragraph reads like
    this The value of an irrational number pi
    is determined from the ancient times up to the modern super computer era. In
    this paper author Late Shri Shantaram Bapurao Janorkar has conjunctured the
    rational number , Christened “GOBA”
    by introducing some concept with references to a circle is nomenclature is like
    π (Pi), but it is not an irrational number. Its value is computed as 3.141592653

    If you look at this research with the
    unselfish vision of a scholar, scientist (researcher) and if you carefully read
    the work published in this publication, you will easily understand the research
    work and ways for further research on this work can be found. In this research
    many new theorems have been established, different new methods have been found
    and in the same way many new theorems and methods will be found. I have made a
    deep study on this research and while deciding on which methods should be
    included in this paper and which methods are easier, I have came across some
    theorems in addition to this theorem and they follow as: For example,

    ® The Theorem of the Evolution or Creation of the Cosmos and Speed. (Wishavachi uttapathi kinva nirmiti vaa vegacha shidhanta, (In
    Marathi).

    ® The theorem of the Evolution or
    Creation of the Radius, straight radius, arc radius and
    diameter of Circle and the
    Circumference of Circle. (Wartul vaa wartul
    parighachi thrigha, saral thrigha, kauns thrigha Aani wyasacha uttapathi kinva
    nirmiti cha shidhanta, (In Marathi).

    ® The Theorem of the total number of planets-stars,
    solar systems and galaxies in the cosmos. (Wishava madhil
    graha-tare, Surya mala Aani Aakashagaangan
    – cha shidhanta, (In Marathi).

    ® The Theorem of the Extent of the Cosmos. (Wishavacha vyapacha shidhanta, (In Marathi).

    ® The Theorem of the definite Period taken by the Creation of the entire
    Cosmos (Saampurna
    Wishava nirmitila lagalelya nishyit welyecha shidhanta, (In Marathi).

    ® The Theorem of the Extent of the Black Hole in the Centre of Galaxies. (Aakashagaangan
    chya Madhya bhagi Aaslelya krushan vivaracha vyapacha shidhanta, (In Marathi).

    ® The Theorem of the Mental Speed (Manachya vegacha shidhanta,
    (In Marathi).

    ® The Theorems of the Numbers, Vedic 16
    Vowels, Vedic 36 Consonants and Their Symbolic Signs. (Aanka,
    vaidik 16 Swar, vaidik 36 vyanjane vaa thyanchya syanketik chinhyancha
    shidhanta, (In Marathi). With this,

    ® Squarization of a Circle = 5605 x 5605 = 31416025 / 10000000
    = 3.14160250 (Wartulache chaurasikaran = 5605 x 5605 = 31416025/10000000 = 3.14160250,
    (In Marathi). In spirituality on the basis of mathematical rules,

    ® Om – The Scientific Verification of Om
    and its Aspect. (Om – Om Che Wiadhyanik Drustya Shidhatha Wa Swarup, (In
    Marathi).

    ® Soul-The Verification of the Existence of Soul, Its Aspect and
    Nomenclature. (Aatmaa – Aatmaaya Chya Asthitwachi Shidhatha, Swarup Wa Namkaran,
    (In Marathi).

    I
    will try to publish in the next issue the further papers on this theorem. Hon.Mr.Dr.T.M.Karade,
    Prof.Dr.Shriram B.Patil, Prof.Dr.Kishor S.Adhav, Prof.Dr.S.D.Katore, Prof.Mr. D.T.Solanke have been
    regularly providing valuable guidance to me and therefore, I am very much
    grateful to them and I shall remain grateful to them forever. In case you find
    any difficulty in understanding the research papers published in this journals
    please send your queries in writing on the editorial address, I shall sincerely
    strive to solve them. You can contact me round the clock on phone or in person.
    This journal is being published by Om Publications, Mahan and it is ISSN, ISBN,
    RNI and ISO 9001:2008 certified.

    All the people, schools, colleges, educational institutes,
    educational boards, mathematical institutes, scientific institutes,
    universities, and the universities listed in the QS (Quacquarelli Symonds) scholars, scientists, researchers all over the world
    can subscribe annually free of charge from the web site http://www.sbjanorkar.com. You can download the e-copy of this international journal in
    PDF format from the web site http://www.sbjanorkar.com and I sincerely
    request you cooperate me in sharing it with your friends, teachers, students,
    schools, colleges, education institutes, education boards, mathematical
    institutes, scientific institutes, all the students of an university and people
    through your email as well as social sites such as Facebook, Google Plus,
    WhatsUp, Twitter, LinkedIn. I dedicate the first issue of this journal to the memory of my
    father and the researcher of this entire work, late Mr. Shantaram Bapurao
    Janorkar, the valuable research done by him, his trust in education.

    ® Mr.Dhananjay S.Janorkar (Chief Editor)

    The
    Function of Publication and Dedication to the World of the International
    Journal through Organization of Shantaram Janorkar Foundation of Mathamatics,
    Mahan

    On the occasion of the birth Anniversary of Late Mr. Shantaram
    Bapurao Janorkar, an International Journal of Shantaram Janorkar Foundation of
    Mathematics, Science and Spiritual, was published through Organization of
    Shantaram Janorkar Foundation of Mathamatics, Village MAHAN – 444 405 Tq. Barshitakli
    Dist. Akola, (Maharashtra State), India. On 15/09/2015 at evening 5.00 P.m. at
    Mahan at the hands of Hon. Professor Dr. Kishor Aadhav, D.Sc., (Head,
    Department of Mathematics, Sant Gadage Baba Amaravati University, Amaravati)
    and dedication of this International Journal to the World was performed at the
    hands of Hon. Professor Dr. S. D. Katore, (Chairman, Board of Studies in
    Mathematics, Sant Gadage Baba Amaravati University, Amaravati), Chief Guest was
    Hon. Professor Mr. D. T. Solanke, (Department of Mathematics, Sudhakar Naik
    & Umashankar Khetan College, Akola), The President of this function was Mr.
    Dhananjay S. Janorkar, (Founder President, Shantaram Janorkar Foundation of
    Mathamatics).

    Along with this international journal, Mr. Dhananjay Shantaram
    Janorkar has determined 504,866,505,707,712,000 (Mile)³/Second The Extent of solar system and
    160,704,000,000,000,000 (Mile)³/Second as diameter as per the principle of
    solar system invented by them. Along with this, numerous principle invention
    papers has been kept before the world through this International Journal and to
    investigate on this invention Mr. Dhananjay Shantaram Janorkar has created a
    new rostrum for global scholars like Ph. D. holders in respect of the same. The
    said international journal/Books can be downloaded free of cost at http://www.sbjanorkar.com
    on this web site. In the said programme Deputy Sarpanch of Mahan, Mr. Pramod
    Bhadange, Smt. Sulbhabai Shantaram Janorkar, Ex-Chairman, Panchayat Samati
    Barshitakli also Secretary of organization Mrs. Jija Dhananjay Janorkar, Mr.
    Dilip Lande Guruji, Mr. Sanjay Ambarkar, Mr. Prabhakar Janorkar, Mr. Uday
    Janorkar, Mr. Suvarnesh Janorkar, Dr. Nitin Janorkar, Mayur Jangle, Rishikesh
    Janorkar, Jay Janorkar and Janorkar family were present at the function.

    Shantaram Janorkar Foundation of Mathematics,

    Village MAHAN – 444 405 Tq. Barshitakli Dist. Akola

    Head Offices:- C/o R.T.Patil House, Near Saraswati

    Vidyalaya, Nityanand Nagar, Gorakshan Road,

    Akola – 444 404, (Maharashtra State), India.

    Phone (Mob): 09021607450, 09226442256

    E-mail: sjfomindia@gmail.com, ijosjfomss@gmail.com

    http://www.sbjanorkar.com

    Note: Hon. Sir, Please share this all of us.

    Link for Free Online International Journal ISO 9001:2008, ISSN (Print & Online):- http://www.sbjanorkar.com/Gernoul.html

    Link for Free Online Books ISO 9001:2008, ISBN: –

    http://www.sbjanorkar.com/booksd.html

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