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The Gion Shrine Problem: A Solution in Geometry

Eighteenth century Japan was a time of isolation and peace, where education and the arts blossomed. Originally posted before 1749 by an unknown author, the sangaku (mathematical tablet) that became known as the Gion Shrine problem, has puzzled people from all walks of life for more than two centuries. Although solutions have been suggested, some of them require mathematics not known in Japan at the time the Gion Shrine problem was written. The problem is still considered unsolved. Using the geometry of Edo Period Japan, the paper's solution is immersed in the culture of that time frame. As identified in this paper, the problem's author deeply understood both the challenge, and inherent beauty, presented by simplicity.


Introduction
As a light breeze rustles the leaves, listening to the responding clank of wooden chimes recalls the words of Charles Moore: "…breezes loosely captured can connect us with the very edge of the infinite." 1 In our current culture of bright lights dispelling all darkness, it can be a challenge to imagine Japanese culture of more than two centuries agoone that embraced shadows. Isolated from the rest of the world, Japan found peace; and in that peace, all aspects of its culture flourished.
Transporting ourselves to Japan of 1750, we find an intellectual environment focused on the beauty of simplicity. Walking down the dirt road, we approach the shrine at Gion. We are dazzled by the sunlight dancing off its bright red paint, amazed at the shrine's fascinating angles and symmetry. Below the shrine's eaves, the wooden tablets and paper lanterns flutter in the light wind; handcrafted objects made as offerings to Shintō kami, the deities of Japan's native religion. Walking under the long extended roof, entering the shrine produces a feeling of anticipation and depth. There is mystery in every corner -the suspense found in darkness as it enhances the light. "…we find beauty not in the thing itself but in the patterns of shadows, the light and the darkness, that one thing against another creates." 2 Lit only by candles, and light filtering in through periodic paper panels, the shrine is filled with imaginary whispers created by dimness. "The quality that we call beauty,.. must always grow from … our ancestors forced to live in dark rooms, [who] presently came to discover beauty in shadows; ultimately to guide shadows towards beauty's ends." 3 The flicker of the candles is captured and enhanced by the highly polished lacquer and hints of gold in the surroundings.
"The small serves to illustrate beauty as well as, perhaps better than, the great; and the poet takes a cultivated delight in simplicity." 4 Delight is found in the simple aspects of life; in finding the simple forms that exist in complexity. This is the world in which the Gion Shrine problem was created; a world of beauty found in simplicity; and pleasure found in the presence of shadows. It was a time of peace and prosperity.
Beginning in the mid-sixteenth century, trade with Europe brought not only financial gain but also Christian missionaries. Upon becoming Shōgun, Tokugawa Ieyasu implemented a ban against Christianity, forcing the religion to go underground.
By continuing this religious prohibition, Ieyasu's descendants 6 encouraged Confucianism to rise, joining Shintō and Buddhism as Japan's religions. Confucianism's ascent changed the Japanese culture in significant ways. Confucianists subscribe to high levels of education and self-improvement, resulting in increases of educational opportunities throughout Japan. Confucianism encouraged reigning in turbulence and replacing it with calm; it embraced straightforwardness and frankness. Japan's literature and art responded with simplicity fostering directness.
Trade with foreign countries initially continued, as did travel overseas. Becoming shōgun in 1623, Iemitsu, Ieyasu's grandson, initiated the period of isolation, sakoku.
Iemitsu reduced trade to be with the Dutch only. Travel overseas was forbidden as was contact with other Asian countries. Japan became an isolated country.
Under Iemitsu, daimyōs were reduced in number through consolidation instead of conquest. 7 Building an administrative structure in the government added administrative positions. A police state was created enforcing laws uniformly, resulting in stability across the nation. Implemented taxation laws focused on agricultural success, increasing the wealth of the lower social classes. Upon becoming Shōgun, Ieyasu moved Japan's capital from Kyoto to Edo.
Enforced capital visitation was imposed requiring daimyōs to spend every other year in the capital city. In years when the daimyō was not a city resident, he was required to leave family members still in the capital. Forced visitations impacted Japan from multiple perspectives. Travel increased, as did urban development. Thoroughfares required inns, taverns, souvenir shops, etc. to support the traveling daimyōs. Increased travel initiated more road maintenance and bridge repair. Required to support two houses, daimyōs were no longer able to support their samurai troops. The overall growth in prosperity was felt throughout Japanese society.
The Japanese began to view the Shōgun as being "wrapped in virtue." 8

Intellectual Aspects of Edo Period Japan
Sakoku was a peaceful time where the samurai put away their swords. In 1615, Ieyasun issued "Ordinances for Military Families". 9 Samurai were beseeched to give equal time to both military practices and refined pursuits. Encouraged to read the Japanese classics, learn medicine, gain musical skills, increase mathematics and calculation knowledge, the samurai class changed. With salaries based on the fluctuating price of rice, samurai were forced to find additional sources of income.
Finding administrative positions, samurai brought with them their attitudes of disciple and devotion, altering the governmental climate and attitude.
Samurai founded military academies, became artists, literary figures, and mathematicians. Matsuō Bashō (1644-1694), one of the greatest haiku poets who has ever lived, was from samurai descent.
In the absence of universities, many samurai opened schools that taught reading, writing and arithmetic. With certification, samurai opened mathematical academies.
Initially restricted to other samurai only, the mathematics academies were later opened to citizens of all class levels. As urbanization and wealth grew, merchants and artisans earned new status. Their burgeoning businesses required these tradesmen to increase their levels of education overall, including mathematics. As the Tokugawa Period continued, mathematical education cut across social levels.
Education actually became a commodity of value that could be compared to a beautiful brocade. 10

Edo Period Literature and Arts
All of the arts flourished during the Edo Period. More leisure time and higher levels of education, increased the demand for literature of all types. This was particularly true with a form of poetry called haiku. Written by Japanese in a single column, haiku uses "cut words" to create pauses when the poem is read. Although there was some variation, the structure was to have a syllable structure of 5-7-5. This means that the first phrase should have 5 syllables when read, the second phrase, 7 syllables, and the last phrase is to contain 5 syllables. As English is a more condensed language, 11   Edo Period wasanka knew Euclidean geometry. Both the quadratic equation and Pythagorean theorem were often utilized. Japanese knew squares, multiplicity of roots, 17 along with trigonometry and logarithms. 18 Pi was known. In fact, several Japanese mathematicians during the Edo Period, calculated new values of pi with great levels of correctness and numerous digits. Fractions were freely used although decimals were also known. 19 Two linear equations could be solved. Wasanka knew how to determine area and volume. As discussed later, many of these techniques were used when solving sangaku.
Geometric relationships were freely studied in Japan, but most studies were algebra based. 20 Prior to the Edo Period, mathematics was viewed as a branch of science. Sakoku cultivated a shift toward viewing mathematics as an art. The impact created by visualizing mathematics as an art was one of speciality. Moving away from generality, mathematics became focused on lengthy calculations. During the eighteenth century, wasanka discussed returning to more generalized approaches to mathematics as a whole. "The writers of the 18th and 19th centuries had here and there tried some problems by the geometrical way, however imperfect they might have been." 21 Reviewing Ajima Naonobu's work, experts state that a considerable amount of generalization was accomplished by Ajima's efforts. 22 Although mathematical knowledge in Japan expanded during the Edo Period, the country did not have access to the simultaneous mathematical developments in Europe.
These accomplishments included calculus. Modern sangaku solutions that involve calculus could not have been solutions intended by wasanka creating sangaku.

Sangaku
All of wasan was viewed as an art during sakoku. This is epitomized by sangaku, hung in shrines and temples as offerings to the various gods, or "votive plaques". 23 San is defined as arithmetic or calculations. The term gaku means plaque, 24   Although most sangaku were created by the members of the samurai class, research on sangaku reflects that people from all social levels authored the tablets. 29 Sangaku painted images did not always exactly match the descriptions given for the problem being presented. 30 This is more than likely due to the fact that the sangaku problem author was not necessarily the same person who painted the sangaku image. If they did not feel capable of painting a sangaku design, wasanka would hire an artist to do so.
There are well known instances where the creator of both the sangaku mathematical problem and the sangaku image were the same individual.  The Gion Shrine geometric design is displayed in the middle of Figure 4's second row. Although the Gion Shrine design is simple, and in context similar to that of the others, it has an asymmetrical balance and grace that sets it apart.
Labels in the sangaku figure and relevant unknowns were identified using characters from the Chinese calendar. 33 Today's mathematics uses characters from both the English and ancient Greek alphabets in a manner similar to that of calendar characters on sangaku.
A typical sangaku had three parts. 34 These three aspects will be used in section have solutions located separately from the problems, challenging readers to try working the sangaku problems themselves.

Sangaku Translation
Reflecting on the historical mindedness of sakoku, sangaku were written in a cryptic language, kanbun, closely related to ancient literary Chinese. Other than diagrammatical marks indicating that it is Japanese, kanbun is essentially Chinese characters following Chinese grammar. Kanbun can be compared to the use of Latin in some older European mathematical texts, such as those written by Newton and Euler.
Considered by Japanese society as an indication of higher levels of education, Samurai were taught kanbun at an early age. Rosalie Hosking mentions the following in reference to the language used on sangaku: "The language of sangaku is quite conceptual, with often just the core nouns and verbs present…. Because of its conceptual nature, the language of sangaku can come across as vague." 38 A different more explicit language was used in texts written for merchants, famers and artisans. 39 In his book, Unger also discusses wasanka's obsessiveness over writing sangaku problems in as few characters as possible. 40 This fits well with sakoku where succinctness was valued not only in writing mathematics but also in literary works such as haiku.  Technique by means of ko circle diameter 2 and otsu circle diameter subtract otsu circle diameter divide square ko circle multiply obtain.

Chapter 3 The Gion Shrine Problem
One sangaku, called the Gion Shrine problem, has evaded solution for at least years. 42 The original Gion Shrine sangaku is long gone due to the passage of time. The original sangaku went undocumented, with its author unidentified.

The Gion Shrine Sangaku History
In

The Gion Shrine Sangaku
With its beautiful balance and elegance, the Gion Shrine sangaku has received significant attention. For more than years, people from diverse walks of life have offered solutions.

Modern Interpretations: The Problem -The Design
Modern documented images of the Gion Shrine sangaku vary as does the statement of the sangaku's problem description and statement of the formulae. Figure   5 reflects four reproductions of modern images that have been published for the Gion Shrine sangaku.
To create the reproductions in Figure 5 as closely as possible to each modern rendition of the Gion Shrine sangaku design, a strict procedure was followed. Copies of each modern image were obtained from either the printed documents or the web. These increase accuracy of each reproduction to its matching modern day image. Once the reproduction was completed, the modern copy was deleted from PowerPoint. Slight differences might be found in fonts used for labels and label placement on the reproduction.
Notice the differences between the images displayed in Figure 5. There are differences in curvature, labels used and label placement. All of these images have a capital "A" and a capital "B" label, although these are only mentioned in the problem's description when defining the length of chord .
Notice that three of the images in Figure 5 have radii and one image has a diameter with a clear center point. Also note that the placement of the radii varies significantly, making a difference in visual beauty and balance. Three different angle measures are represented by the radii and diameter of the images shown in Figure 5.

Modern Interpretations: The Problem Description
In the modern presentations of the Gion Shrine sangaku, there are also differences in the statements of problem description. Utilizing the same sources as those in Figure 5, following are the authors' problem descriptions.

Modern Interpretations: The Answer
The first solution sangaku to the Gion Shrine problem was posted years ago.
Including Tsuda's initial solution sangaku, no one offering a solution has mentioned that the original sangaku, hung prior to 1749, included a numerical answer. This includes the four modern sources being compared in this paper. It can be safely concluded that no answer was given on the original tablet.

Modern Interpretations: The Formulae
In the four modern presentations of the Gion Shrine sangaku, there are also differences in the statement of the formulae. The modern sources discussed previously are compared following.  It would also be reasonable to speculate that Tsuda, the first individual to post a Gion Shrine problem solution, might have made the same ka assumption, using "+" symbols where none were intended by the original author.

The Gion Shrine Sangaku: My Interpretation
My interpretation will follow the structure usually provided by sangaku themselves. First, I will focus on my interpretation of the Gion Shrine sangaku problem as shown by its design and the accompanying design description. As it has been discussed previously, it is evident that no answer was provided on the original Gion Shrine sangaku, so this portion will be omitted.   In Figure 6, note that = 2 + 2 .
Although I utilize the diameter as shown by de Reyna and Clark, I conclude that the use of in the formulae is correct. The formulae closest to fitting with my solution is given following; although I conclude that using a strict mathematical interpretation is not correct.
"Using = + + + and = + + , express , , , and in terms of and ." Following, I restate the sentiment of the formulae that I believe was actually written on the sangaku containing the design in Figure 6. It is important to keep in mind that kanbun is a cryptic language, and one that I do not know. My interpretation here is in modern language with an intent of conveying a general meaning only.
Written on the tablet in kanbun characters probably were lines similar to the following: "Put is and and and " for " is related to and related to and related to "; "Express , , , and with and ." It would be very natural for mathematicians, such as ourselves, to read these lines and immediately translate them into mathematical meaning and mathematical symbols, especially if they were written on a sangaku.
There is much discussion when it comes to the translation of historical

Chapter 4 The Solution in Geometry
The variation of the images displayed in Figure 5 indicates the difficulty that exists in translating mathematics that belongs in the past. This is especially true when the original sangaku no longer exists. Each interpretation is open to each author's own viewpoint. As mathematicians, we bring our entire self to any problem that we address.
Although mathematics insists on a high level of precision and accuracy for it to have the greatest meaning, the fact that we each have different viewpoints can be of significant benefit to our field of study. We learned long ago that problems are often solved more completely with the synergy provided by multiple individuals, especially when addressing complex problems. However, variation in interpretations creates uncertainty in knowing truth. A balance between the two situations must be reached. Comparing the Gion Shrine sangaku design presented in this paper, Figure 6, and the four modern design interpretations in Figure 5, which image is closest to the original sangaku image?
It would be a natural inclination for wasanka to assume that the Gion Shrine sangaku was like all of the rest. I feel that a purely mathematical meaning was not intended and translating the kanbun characters on the Gion Shrine sangaku into mathematical symbols is making a false assumption that produces a misintended result.
The characters painted on the Gion Shrine sangaku were to be interpreted just as they were written, not as mathematical statements, requiring mathematical symbols.

My suggested statements for and creates a single sentence, "Put is and
and and for is related to and related to and related to ". This is a different manner of interpretation than prior mathematicians. My concept for the solution utilizes and to express the various pieces of the sangaku design. This emphasizes the last formulae statement, "Express , , , and with and .", that I suggest.
The characters used as and were meant to represent clues to a mathematical   Figure 7). The angle between the two red lines is the central angle for the circle sector, part of which is the Gion Shrine sangaku design's circle segment.
As wasanka would have done, using a protractor, the interior angle of the circle sector in Figure 7 is found to be 140°. The radius of the large circle, is equal to 2 . With = 2 + 2 , then the formulae for the perimeter of the circle segment relates and as indicated by the second relationship mentioned in statement .
The last relationship given with clue , is " related to ". This relationship indicates the first and the last lines of the mathematical haiku. Table 2 contains the haiku, in both the sangaku's geometric shapes and the geometry that goes with each shape.

Gion Shrine Perimeter and Circumference Haiku
The simple geometric beauty of this solution is as magical as the sangaku design.
For reference purposes, the image from Figure 6 is reproduced here. Examining the geometric expressions given, the haiku syllable pattern of 5-7-5 is followed with the middle line longer than the other two. The first line reflects 2 . In the design shown in Figure 6, the following relationship applies to chord : Looking at statement given in my sangaku formulae, the first relationship is related to . Based on = 2 + 2 , the variable that shares with is .
The geometry on the second line is 2 + 2 + also the name of the peninsula northwest of Kyoto, where it is believed his mother was born.
His loving humor was well known. It has been suggested that the following haiku

Justification
As mathematicians, surmising is not considered proof, especially when it is in connection to a mathematical problem that has gone unsolved for more than two and a half centuries. With historical documentation often being absent, and in this specific case, the original problem (the original sangaku) actually missing, there are some extrapolations that must be made. Many of the suppositions presented here are based on experts' interpretations and on as much documentation as was available close to when the Gion Shrine sangaku was written.

Justification of Solution
There are three pieces to the justification of the solution presented. The first rests in probable discussions occurring amongst Japanese mathematicians at the time that the Gion Shrine problem would have been inked on its wooden tablet. The second is focused on translation of ancient texts, and the variation of even current translations.
Unification of image, the solution, and Japanese culture at the time is the third justification.
There is evidence that around the mid-eighteenth century, Japanese mathematicians began to discuss the need for wasan to move in a direction of generality. As mentioned in section 2.4, "Wasan -Japanese Mathematics", Ajima firmly  75 Hosking also amplifies the fact that sangaku were created with a definite link between the image present and the accompanying text.
As pointed out by Hosking, this is particularly true when neither the problem description or answer have numerical values, as is the case with the Gion Shrine sangaku.
Incorrectly translating the formulae on a sangaku leads to false conclusions. Partly due to a translation error, the entire meaning of the Gion Shrine sangaku was missed for hundreds of years.
Sakoku was a unique period of Japanese culture. Education of all types was encouraged allowing it to cut across social levels. While Japanese mathematics became specialized, other aspects of the Japanese culture found a rich environment for development. With the embracement of Confucianism, a shift occurred in the general cultural personality towards directness. The simplicity of haiku folded perfectly into this cultural shift.
Designed to be open-ended yet direct, haiku turned towards mathematics in the midst of sakoku fits profoundly well from a cultural standpoint. The concept of a mathematical haiku corresponds beautifully with the Edo Period culture.
The Gion Shrine sangaku was intended to be different not only in design but also in its solution; a solution that fits into every aspect of mid-eighteenth century Japanese culture.  Mastering the art of creating haiku requires a precision of mental ability that is extreme. Haiku poets must drill down to the essence of ideas, images and situations.

Justification of the Sangaku Author
They must be able to construct and convey complete ideas in seventeen syllables.
Only the mind of an artist would visualize a distinct asymmetric design that had such perfect balance and grace as that of the Gion Shrine sangaku image. An artist would understand the need for related proportions in that same simple balanced design so that it would evoke a feeling of beauty. who was the jealous "thorn bush". These are suppositions, but similar to suppositions made by Buson scholars.
On the fiftieth anniversary of his mother's death, his nostalgia was documented through his own verse. It is certainly reasonable that he might have commemorated his mother's death at its twentieth anniversary, in 1748. Buson would have been close to 30 years old. The memory of his mother's passing would have still been in his mind, along with possible bitterness that he felt towards his father, and possibly towards mathematics itself.
By documenting the conversation that he had with his disciple Shōhu, Buson knew that the advice would be conveyed to all future haiku poets, and others such as ourselves. The message to be heard was to treat life as a banquet with every delectable relished with delight. He believed that each intellectual discipline gains from the others.
Mathematics profits if wasanka try writing haiku or creating philosophy or charcoaling a drawing. Only then will you master your technique. Hanging a tablet with a mathematical problem solved only by thinking outside of mathematics itself, Buson crafted an image that evokes balance and grace; an image accompanied by a beautiful solution. Buson was sharing his art, his poetry, and his beliefs. Buson was a genius haiku poet, a master artist, and a brilliant mathematician.
Buson's message, '…stroll among the trees in the garden, hold a banquet by a mountain stream, and enjoy a conversation over sake wine. … the fragrance of blossoms will come drifting in the breeze, and the image of the moon will be seen floating on the water" 83 can still be heard today.

Sangaku Author's Message
Mingling in Kyoto's cultural circles, Buson overhears several mathematicians discussing the need for mathematics to become more general in method and he embraces that thought. As the twentieth anniversary of his mother's passing approaches, he develops an idea that fits perfectly with his innate desire for the various intellectual disciplines to share their knowledge; for the disciplines' practitioners to learn from other aspects of intellectual pursuits. The wasanka, so focused on the existence of there being only one way to solve the sangaku moving in the breeze, forgot about the air itself. Awaiting discovery by the wasanka, there was another solution to the Gion Shrine sangaku; a solution that needed only a few numbers; a solution that rested in beauty; a solution that ends in simplicity.
Buson's message was one to be heard for generations. Einstein often suggested that everything should be challenged. With his thought experiments, he imagined the impossible. We all gained from his strange imaginings.
The past aides us well by setting a foundation to learn from in creating a new future.
It is important to find the simple forms that exist in complexity.
Imagine the impossible, and make it real.