#mathematikoi

[alumbradoss]

Beyond the Veil: Spacetime and Frequency Domain

“Have you ever pondered what lies beyond the familiar confines of spacetime? The concept of the frequency domain serves as an enigmatic boundary to our known universe. Imagine reaching the speed of light – such an event could theoretically catapult you out of spacetime and into the mysterious frequency domain. This isn’t just science fiction; it’s a gateway to a realm where the rules as we know…

[pcfunlab]

Top 10 Contributions of Pythagoras.

01.Mathematics

Pythagoras started the idea of a numerical system, and therefore the beginning of mathematics. To the Pythagoreans, genuine numbers were the most vital thing, and numbers make up the world.

02.Pythagoras’ Theorem

Pythagoras is most famous for his ideas in geometry. He was the first to propose that the square of the hypotenuse (the side of the triangle opposite to the right angle) is equivalent to the sum of the squares of the opposite two sides.Even though this hypothesis was first put forward by the Babylonians, Pythagoras was first to demonstrate it. It is additionally thought that he invented the tetractys, a triangular figure consisting of 10 points arranged in four rows, with one, two, three, and four points in each row. Pythagoras believed that 10 was the ideal number.

03.Pythagoras’ Philosophy

Pythagoras established the mysterious society of the Pythagoreans in southern Italy. The Pythagoreans put forward a hypothesis that stated that everything known to humankind could be clarified with numbers, explicitly entire numbers. This rationale made perfect sense to them, and it is easy to see why. Even today, we use numbers for everything from working out wind speeds in a storm to calculating the speed of a vehicle.

04.Conclusion

Pythagoras made huge contributions in the fields of astronomy, mathematics, philosophy, music and many other areas. He is considered the “father of philosophy,” and this can be seen through his teachings, his theories, and his philosophies. His impact on later philosophers such as Plato cannot be underestimated, and his influence was significant to the point that he could be seen as the most persuasive intellectual of all time.

05.Early Christianity

The historian Eusebius compares Pythagoras to Moses, while Augustine of Hippo (354–430 AD), the early Christian theologian and philosopher, dismissed Pythagoras’ theory of metempsychosis without actually naming him, yet generally communicated profound respect for him. In On the Trinity, Augustine discusses Pythagoras’ modesty in referring to himself as an “admirer of insight” rather than a “sage.”

At around a similar time, Neopythagoreanism started to become more prominent, and the first century Neopythagorean logician Moderatus of Gades was instrumental in developing Pythagorean number philosophy and categorizing the spirit as a “sort of numerical harmony.”

06.Greek Reasoning

Aristotle states that Plato’s reasoning was heavily reliant on the lessons of the Pythagoreans. Plato may also have obtained the idea that mathematical and dynamic ideas are behind logic, science, and morality from Pythagoras. Plato and Pythagoras shared a magical way to deal with the spirit and its place in the material world, and it is likely that both were impacted by Orphism, a set of religious beliefs and practices originating in the ancient Greek world.

07. Religious Lessons

Pythagoras’ religious lessons were based on the principle of metempsychosis, which states that the spirit never dies and is bound to a cycle of reincarnation until it can free itself from this through virtue.

A soul was believed to exist in both animal and vegetable life, even though there is no proof to indicate that Pythagoras thought the spirit could be contained in a plant. It could, however, be contained in the body of a creature, and Pythagoras professed to have heard the voice of a dead companion in the wail of a dog being beaten.

08. Cosmology

Pythagoras was the first person to suggest that the earth was a sphere, but it is not clear what led him to that conclusion. It is possibly connected to his belief that circles were the strongest shape.

His experience of the universe was most likely exceptionally basic: at that time, the earth was still thought to be the focal point of the universe with everything revolving around it. The Pythagorean view of the universe was pretty straightforward and did not take into account any observation of the movements of the planets. They held the belief that the planets all moved in giant circles, and that when they brushed against one another, they made a sound. They believed that these sounds were melodic harmonies, and this music has come to be known as the Music of the Spheres. This music was not audible because it was a constant background noise.

The Pythagoreans also held the belief that the earth, planets, and stars all circled a central flame, and night and day was caused because of this movement. They believed that a “counter earth” existed on the other side of the flame. Since they felt that fire was more important than earth, the focal point of the universe must be fire.

Pythagoras was the first to perceive that Venus at night and Venus toward the beginning of the day were the same planet.

09.Music and Lifestyle

The pursuit of music may have been associated with the love of Apollo. The Pythagoreans believed that music was a purification for the spirit and had a similar effect to that which medicine produced on the body. One of Pythagoras’ stories reports that when he witnessed some inebriated youths attempting to break into a lady’s home, he sang to them, and the young men’s “furious persistence” was silenced.

Pythagoras was the first person to recommend music as a prescription. He connected music to craftsmanship, design, government, raising a family, fellowship, and self-improvement. He thought it was possible to align the souls to their perfect nature, and through music he performed what he called “soul adjustments.” Pythagoras also associated arithmetic with music and believed that music should not be seen as a diversion alone. He believed that music was an outflow of harmonia, the divine rule seeking to banish confusion and conflict in the cosmos. Along these lines, music was seen to have a double function as, like science, it empowered people to see into the structures of nature.

The Pythagoreans also set great store by physical exercise and recommended daily morning walks and sporting activities. Periods of self-examination at the beginning and end of every day were likewise advised.

10.The Public Way of Life

Both Plato and the ancient orator Isocrates suggest that Pythagoras was an advocate for a new way of life. The school, or organization, that Pythagoras established at Croton operated much like a monastery, and its members shared their assets in much the same way. They were also committed to one another to the exclusion of outsiders. One Pythagorean saying was koinà tà phílōn which meant “All things in like manner among friends.”

There were two groups within Pythagoreanism: the mathematikoi (students) and the akousmatikoi (listeners). The akousmatikoi were generally regarded as “old masters” in enchantment, numerology, and religious teachings, while the mathematikoi were recognized as an innovative group which was more progressive, pragmatic, and scientific.

[waywardpencil]

PYTHAGORAS

WEEK 2 [Day 3]

PYTHAGORAS the first "true" mathematician

Pythagoras established a school in southern Italy around 530 BCE,  and it harbored the most bizarre and mystical beliefs regarding math and religion that were imposed onto their worshipers students.

strict vegetarianism

never urinating towards the sun

never marrying a woman who wears gold jewelry

never passing an ass lying in the street

never eating or even touching black fava beans

and loads of other odd rules.

The members were divided into the "mathematikoi" (or "learners"), who  developed the mathematical and scientific work, and the "akousmatikoi" (or "listeners"), who focused on the more religious and ritualistic aspects. There was always a certain amount of friction between the two groups and eventually broke up and got segregated.

People despised the Pythagoreans, their hate and resentment towards the secrecy and exclusiveness of the group exploded like a ticking time bomb and ended in a massacre, in 460 BCE, all their meeting places were burned and destroyed.

The main dictum of Pythagoras's school was “All is number” or “God is number”

The Pythagoreans effectively practiced a kind of  number-worship, and considered each number to have its own character and meaning.

The number one was the generator of all numbers

Two represented opinion

Three, harmony

Four, justice

Five, marriage

Six, creation

Seven, the seven planets or “wandering stars”.

Odd numbers were thought of as female

Even numbers as male.

[scottbenzel]

Mathesis and Mathematikoi, Knowledges 2017, curated by Christina Ondrus, Mt. Wilson Observatory, CA

[universallyladybear]

De la terre et le ciel étoilé sous le dôme du planétarium une séance contée et interactive qui permet de découvrir une autre planète…

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Que les continents étaient probablement en place très tôt après la formation de la nébuleuse se constituent les premiers corps solides les planétésimaux embryons des futures. De manière partiellement liquide aujourd’hui plus de 80 de la terre au soleil mais la distance de jupiter au soleil et une octave du soleil jusqu’au ciel. Saturne et les anneaux et les sons célestes fussent-ils loin ou près[14 de façon plus large encore certains auteurs pensent à l’harmonie cosmique dans le gorgias 507e platon parle des sages qui.

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Youtube Les Planètes De la terre et le ciel étoilé sous le dôme du planétarium une séance contée et interactive qui permet de découvrir une autre planète...

[husheduphistory]

Pythagoras: The Mystic Madman of Mathematics

It is a subject despised by many. For those who detest it, stomachs churn and heads spin in an arena where numbers run rampant. Sometimes, letters get involved in a free for all of mind-bending mayhem. 

Math.

Throughout the numerical battle there are a few snippets that stay entrenched in the brain and one of these is the Pythagorean theorem. Regardless if anyone actually uses it in daily life, the combination of  A2+B2=C2 (A squared plus B squared equals C squared) remains familiar.  Although the theorem may be hammered hard into the heads of students every year, the person behind it is rarely discussed in math class. The reason for this may be because the man was even more mind boggling than his namesake. Born in approximately 570 B.C.E. in the region of Samos, Greece, Pythagoras was able to travel extensively with his merchant father which gave him exposure to scholars all over the world beginning at a young age.  In 535 B.C.E. Pythagoras made his way to Egypt where he visited temples and took part in discussions, eventually completing the rites to priesthood. His time there came to an abrupt end when in 525 B.C.E. Cambyses II, the king of Persia, invaded Egypt and when the Egyptian resistance fell Pythagoras was counted among the prisoners of war taken to Babylon. While in Babylon Pythagoras kept company with the priesthood of Magoi (Magi) who instructed him in their rites and mystical teachings until he returned to his homeland of Samos approximately five years later. It was here that Pythagoras formed his first school, called the "semicircle", where his followers gathered to discuss "questions about goodness, justice, and expediency" while being taught by a "man who made all these subjects his business.” When he was not taking part in his group study Pythagoras spent his time in a cave outside the city where he immersed himself in his biggest obsession, mathematics. 

In the time of Pythagoras, philosophy and science coexisted but he believed that at the core the entire meaning and structure of the universe lay in the combination and manipulation of numbers. These were not mere figures to Pythagoras, he believed that mathematics were the very embodiment of divinity. He was not the only one to believe so. Pythagoras eventually settled in Croton, located in southern Italy, and it was here that he formed his second school focusing on the study of sacred numbers under the umbrella of what was to be called Pythagoreanism. 

Bust of Pythagoras at the Capitoline Museum in Rome, Italy.

Pythagoreanism was deeply rooted in the transmigration of soul and the belief that the soul was reborn in earthly beings over and over again. This cycle could only be broken based on the purity of the life last lived and an understanding of the functioning of the universe, which to Pythagoras and his followers, called Pythagoreans, lay in the study of mathematics. This society was made up of two groups that included both men and women. One group, called akousmatikoi (hearers) took part in the studies as part of their daily life. To the other group, the mathematikoi (learned), the society was their daily life. The members of this inner circle paid the price of membership by subscribing to a set of extremely strict rules set forth by their leader. All worldly belongings were to be given up, clothing was now restricted to simple garb, sleep was practiced only very lightly, they had to live together within the society, alcohol was absolutely not permitted, and they all had to be strict vegetarians, possibly due to their belief of souls cycling through living beings repeatedly.  These were accompanied by other more bizarre rules such being forbidden to pick up an fallen item, to pet a white rooster, or travel any high roads. 

While many philosophical societies spoke and contemplated divinity in the form of gods, to the Pythagoreans, the entire universe was rooted in numerical harmonies and reality was found in mathematics. These numbers each had their own meaning. The number one was the generator of all numbers, two represented opinion, three stood for harmony, four was for justice, five represented marriage, six was creation, seven stood for the seven planets, and so on. The number ten was a supreme number that could be created by adding one, two, three, and four. This concept could be drawn out by having rows of dots forming a triangle, called a tetractys, a sacred symbol to the Pythagoreans. Odd numbers were thought of as female, even numbers were considered male, and it was to these numbers that the society prayed, swore oaths, and even made sacrifices. Upon their discovery of the Pythagorean theorem, (sometimes represented by three squares and called the 47th Problem of Euclid) the vegetarian society celebrated by sacrificing an ox to the mathematical calculation. In another contrast to their vegetarian lifestyle, followers of Pythagoras were forbidden to eat beans. This was because their leader fully believed the bean was capable of containing a human soul and he likened eating a single bean to "knawing on the heads" of one's parents.

An illustration of the Pythagorean Theorem represented by the three squares as the  47th Problem of Euclid.

French manuscript dated 1512/1514 showing Pythagoras refusing to eat from a bean plant.

Woven into their mathematical calculations were deeply held mystic beliefs about the universe around them. Due to the observation that nothing is without a center, the Pythagoreans considered the circle to be the "parent" of all other shapes. A circle drawn with a point in the center was called a "monad" but it was also referred to as The First, The Essence, The Foundation, Unity, and by Pythagoras himself this symbol was called "god and the good.” A line segment in a geometric shape was called a "dyad" and it was not looked upon well by the Pythagoreans. To them, this line segment represented anguish, separation, and the cruel force standing in the way of the return to oneness. Pythagoras called the dyad "a deity and the evil, surrounding which is the mass of matter." The previously mentioned tetractys may have looked like a simple triangle made from dots but each layer of these marks represented something much deeper to the followers which is why is was so revered by them. The top mark of a single dot represented the point of divine wisdom through which everything was created. The second row of two dots stood for the 'first dimension" of Neikos” or Strife. It was here that the impulses of movement and division resided. The third row of three dots was the "second dimension" of “Philotes” or Harmony where physical beauty and mental balance came together. The final row contained four dots, each of which represented an element of earth, air, fire, and water. This simple looking shape served as a divine map from the heavens to the very ground they stood on. 

The Pythagorean society grew to the size of approximately 2,500, partially due to the fact that Pythagoras would turn no one away from his teachings. Men, women, young, old, poor, rich, anyone was welcome. Those in his company considered themselves incredibly lucky because to the Pythagoreans, Pythagoras was not merely a teacher, he was akin to a god. Stories about their leader swirled and expanded among them stating that he could calm the wind and seas and once guessed the correct number of fish in a net dragged in by fishermen. When he ordered they be returned to the water the fishermen complied and reported that miraculously all of the fish swam away alive. Additionally, his followers reported that Pythagoras had a "golden thigh" literally made from the precious metal. Surely then, he must be greater than human. The only justification they ever needed for anything lay in one phrase. As reported by Roman historian Cicero, any ideas the Pythagoreans had could be defended and validated simply by saying "the master (Pythagoras) said so.”

Pythagoras depicted in Raphael’s fresco The School of Athens.

However, not all remained in favor with their leader. One of the most important rules of this intellectual circle was a commitment to absolute secrecy about the teachings and activities of the group. One of the followers by the name of Hiappasos made the mistake of discussing the properties of the dodecahedron shape with outsiders. For this, he was expelled from the community. This banishment and disrespect toward the rules were unfathomable to some fanatical followers and they went to great lengths to ensure they they would never commit the same offense. One of Pythagoras's followers bit off her own tongue to prevent herself from ever revealing any secrets. 

These fates are minimal though compared to the consequence faced by the man who challenged Pythagoras and his holy numbers. The belief system of the Pythagoreans was based in whole numbers and ratios and one of the followers by the name of Hippasus threw the group's structure into jeopardy when he revealed his discovery of irrational numbers. According to the story, Hippasus was on a boat with Pythagoras and a group of followers when he presented his findings to their leader showing that the square root of two was not a whole number, but a string of never ending numbers. While this was a groundbreaking discovery to the field of mathematics, to Pythagoras and his followers this was nothing short of blasphemous treason. It is said that Pythagoras attacked Hippasus, wrestled him over to the side of the boat and held his head underwater until he fell lifeless. His body was reportedly then shoved overboard and Pythagoras warned his loyal onlookers to never speak of the incident. While the specifics of the story are murky it is known that a Pythagorean fable told by his followers stated that Hippasus was "drowned by the gods" as a punishment for revealing irrational numbers to the world. Whether Hippasus died in a tragic accident or at the hands of his teacher, the message remained the same. Do not reveal our secrets and do not cross Pythagoras. 

Pythagoreans' Hymn to the Rising Sun by Fedor Andreevich Bronnikov, 1869.

As Pythagoras's community grew in size and influence the surrounding city grew more and more uncomfortable. In a time rife with political unrest and mob rule, cults were becoming rampant due to their promise of enlightenment, security, and secret mystical knowledge that made the common person feel superior to the tumult surrounding them. Approximately twenty years after his arrival in Croton, a nobleman named Cylon approached the Pythagoreans asking to join their cult of numbers. Cylon had a unsavory reputation and his rejection by Pythagoras sent him looking for revenge. He went to the masses, who were already displeased with the goings on in their city, and rallied them together to move in on the cult. As Pythagoras and many of his followers sat inside the home of one of his disciples, the angry mob set the structure on fire inciting a massive blaze that quickly consumed the commune. While huge numbers of his followers were killed due to them all living together in close proximity, Pythagoras allegedly escaped only to face death shortly after. 

There are two main stories that tell of the end of Pythagoras. In one version of events the teacher fled the inferno and made his way to Metapontum where he secluded himself and eventually starved to death. Another account tells that Pythagoras died observing his beliefs until the very end. According to various reports, Pythagoras was able to flee the site of his burning compound but on his escape route he suddenly found himself standing before a bean field. With these sacred plants at his feet, separating him from freedom, Pythagoras remained loyal to them and refused to step upon a single bean. The angry mob caught up to him and cut his throat, spilling his blood on the small pods he swore to never harm.

It is from this point at approximately 500 B.C.E. that  Pythagoras disappeared from his strategically built stage with some people claiming to have sightings of him for years to come. Regardless of his final moments and the many strange practices that filled the days before his death the theories and philosophies of Pythagoras and his Pythagoreans led to shifts in thinking and concrete discoveries that help shape the world both then and now. From far reaching ideas about spiritualism and influencing the likes of Plato and Aristotle to fleshing out the field of geometry and appearing in the familiar cluster of A2+B2=C2, the modern world was changed by a brilliant man who lived and died honoring his beliefs....and beans.

[darioalvarez]

"... la existencia, en la primitiva comunidad pitagórica, de dos clases de miembros: los matemáticos(mathematikoi), «conocedores», es decir, a quienes Pitágoras comunicaba los conocimientos científicos y los acusmáticos (akousmatikoi), «oidores» a quienes participaba de los conocimientos, creencias, principios morales, ritos y prescripciones específicas de la hermandad sin exigirles que conocieran en profundidad las razones de su credo y su proceder. ..."

[misswanderings]

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35

I love old photos and their ghostlike capturing of moments and memories. This one of a man standing on the head of a Sphinx in Egypt with two pyramids behind him instantly takes you away from the modern package deals and politically correct holidays the bourgeoisie have today and then discuss over their dinner parties. Clearly if everyone climbed on the Sphinx’s head there would be no Egyptian pyramids to visit, and their beauty could not be marvelled at, Yet I am aware of how so many breathtaking natural and man made sites have been destroyed by tourism. The way hoards of buses and people cram in to see them, with the earnest narratives on their head-sets giving the history and the carefully constructed fences and wheelchair access lanes ushering them in single file down the paths you can take. You name the landmark, I’ve been there and observed it with bus loads of tourists with their selfie sticks and had it destroyed. I suppose though this image shows just why we do need to put up rails, it is the modern instagram photo shot – go to the spot, take photo of me there, claiming it. This is why the lonely austere beauty of Wedding Cake rock in the national park, or the figure eight pools I have visited since I was a child have had to be fenced off, as social media has made these landmarks to claim and project yourself within, on or close to.

  The Egyptian pyramids, build around 2500 BC, were remarkable in their construction and beauty, yet they were not nearly as elaborately designed to mathematical proportions as the Ancient Greek and Roman buildings that were designed with Pythagorean geometry. There is no written evidence of the mathematics used in the construction of the pyramids, yet it does seem they follow very similar designs to the foundations of geometry later employed and notated by the Greeks, particularly with the Great Pyramid of Giza. The Egyptian pyramids were made out of square blocks, which meant that they ended up as square pyramids, rather than the mathematically simpler and geometrically regular tetrahedrons. The Greeks were of course obsessed with symmetry and their architecture and worship of numbers and precise geometrical forms is reflected in their architecture. Indeed beauty to the Greeks needed to have a triad of symmetry, proportion and harmony, and their need to reflect these values in their lives, led to the creation of many important mathematical concepts that have been attributed traditionally to Pythagoras (570-495 BC), though in reality the precursors of these ideas were no doubt based on earlier ideas of the Egyptians and Babylonians. Some of the most important ideas of the Greeks when it comes to harmony, beauty and proportion are the concepts of The Golden Rule, The Pythagorean Theorem and the mathematical study of figural numbers.

  A figural number is one that can be expressed or represented by a geometrical arrangement of equally spaced points (pebbles or spheres). Pythagoras and his followers have usually been attributed with the creation of generating figurate numbers using gnomons, though the evidence that survives is rather scant on this. The gnomon is the piece that needs to be added to a figurate number to transform it to the next bigger one. For example, the gnomon of the square number is the odd number of the general form 2n + 1, n = 1, 2, 3, ... These formulas allows explicit formulas for all triangular numbers that are also perfect squares.

  35 is a figural number and it is a tetrahedral number, or triangular pyramidal number.). So 35, if represented by 35 equally spaced points, will form a pyramid with a triangular base and three sides, this is called a tetrahedron. A triangular pyramid with a side length of 5 contains 35 spheres, each one represents one of the first 5 triangular numbers. The tetrahedron is the simplest polyhedron with four our faces and six edges. All the faces are triangles. If they are all equilateral, such a tetrahedron is called regular. Accordingly, the tetrahedron is both a right triangular pyramid, the base of which can be any of the faces. You can build a tetrahedron with layers of spheres. The number of the spheres in one layer is 1,3,6,10..., generally n(n+1)/2.  If you add the spheres layer by layer, you get the tetrahedral numbers 1,4,10,20,..., generally 1+3+6+10+...+n(n+1)/2=n(n+1)(n+2)/6. The tetrahedron is widely used in the art and prevalent in the wild: fruits of some plants in racemes with the exact shape. A tetrahedron is one of the five Platonic solids that are sees as ideal primal modes of crystal patterns that occur throughout the world of minerals. The Greeks believed this solid represented fire.

  The Greeks believed that harmony passed from an arithmetical concept of number to a spatio-geometrical concept of ratio. In some languages tetragonal became a figurative term for someone of steadfast character.The tetraktys, or the figural decad was the most important figural number, it was a symbolic figure to which the Pythagoreans swore oaths; it represented a perfect reduction of the numerical to the spatial, strength, justice and perfect equality. A tetraktys was a perfect triangle made of ten points, and the Pythagorean oath was sworn on it:

          By that pure, holy, four lettered name on high,

          nature's eternal fountain and supply,

          the parent of all souls that living be,

          by him, with faith find oath, I swear to thee."

The Pythagoren religion of worship around the ten object triangular number, the tetractys. Full members of the society were called the mathematikoi (which is the origin of our word “mathematics”); those outside the inner circle of full members were the akousmatikoi (the origin of our word for acoustics). These could listen in on the discussions of the mathematikoi, but could not themselves take part. It is also often asserted that many of the discoveries made by the society were in fact joint efforts, but were all respectfully attributed to Pythagoras himself. One of the most central mathematical concepts and the foundation of aesthetics is sometimes attributed to Pythagoras’ wife Theano. Theano who was born in Crotona c. 546 BCE, wrote a treatise describing the 'Golden Mean" that is the first clear reference to what has become known as the ‘golden ratio’ or the ‘golden section’.

  According to legend Pythagoras discovered the concept the golden mean when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers. Theano was very interested in these ideas and when Pythagoras came to Samos, she went to hear him speak. She and Pythagoras married although she was 36 years his junior. They had 5 children and she also wrote a number of his ideas into books, the most central one was the golden mean which he saw as the ratio that exists within flower petals, trees, seeds, natural foods, shells, the galaxies in the universe; a mathematical language of the universe.The Pythagoreans endeavoured to keep this language a secret; declaring that anybody that broached the secret would get the death penalty. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance.  The symbol of the Pythagorean brotherhood was the pentagram, in itself embodying several Golden Means.

    The Golden Ratio exists when a line is divided into two parts and the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which both equal 1.618033988749894848204586834365638117720309180, now known as Φ. However it wasn’t until the 1900’s that American mathematician Mark Barr used the Greek letter phi (Φ) to designate this proportion. The golden ratio is a unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. As such, it symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage. In nature, we find patterns, designs and structures that follow geometrical archetypes. These relationships were studied famously by Leonardo Da Vinci and become fundamental to renaissance thinking. The Golden Ratio reveals the underlying metaphysical principle of interconnection between all things, the inseparable relationship of the part to the whole; that ‘everything impinges upon everything else’. These ideas were expanded fully In Luca Paciolo’s book whose title references the golden ration, De divina proportione (On the Divine Proportion). The book is illustrated by Da Vinci and composed around 1498. Its subject was mathematical proportions and their applications to geometry, visual art, and architecture. The clarity of the written material and beauty of the diagrams helped the book to achieve an impact beyond mathematical circles and bring Pythagorean geometry into the wider discourse. By this time this ubiquitous proportion was known as the golden mean, golden section and golden ratio as well as the Divine proportion.

  If it wasn’t for Theano the golden mean which permeates the architecture of all forms may not have been attributed to Pythagoras. After his death Theano became the head of Pythagoras' school and, with the help of her daughters,(Damo, Myria and Arignote) all of whom were philosophers and one of her sons, she continued the Pythagorean school of wisdom. She and her children not only kept the school and its doctrines alive they were central to the spread of Pythagorean thought. Without Pythagorean thought some of the most central elements of Western philosophy based on the Golden Mean would not exist. As Aristotle later put it, the Golden Mean is the Greek ideal of the desirable middle between two extremes, one of excess and the other of deficiency.

[alumbradoss]

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[alumbradoss]

#synchronicity #Love #Gnosis #Math #Awakening #Magnetism #Manifestation #Knowledge #Mathematikoi

[alumbradoss]

#knowthyself #Illuminism #Wisdom #Knowledge #numbers #Gnosis #God #Gnosticism #enlightenment #enlightened #Mystics #Spirtuality #Reality #awakening #Indigo #InitiatesOfTheFlame #Starseeds #Otherkins #Monad #ScientificIlluminism #HyperHumans #Mathematikoi Time to ignite the Spark.

[alumbradoss]

“Philosophy and Nature is written in that great book which ever is before our eyes –I mean the universe –but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.” - Galileo #Math #illuminism #Labyrinth #Initiation #Mind #Nature #Reality #Symbols #Geometry #Universe #hyperhumanity #hyperhuman #projectfallenstar #mystery #secret #evolvedmind #truth #evolution #philosophy #logos #mathematikoi #language #path #consciousness #Seeker