The Sacred Circle


Link to Squaring the Circle post

Link to Pi and Infinity post


Link to Potential Geometry Post

LINK to Sacred Geometry post



review videos on euler’s identity

A study of the brains of sixteen mathematicians found that the “emotional brain” (specifically, the medial orbitofrontal cortex, which lights up for beautiful music, poetry, pictures, etc.) lit up more consistently for Euler’s identity than for any other formula.

Zeki, S.; Romaya1, J. P.; Benincasa, D. M. T.; Atiyah, M. F. (2014), “The experience of mathematical beauty and its neural correlates”  Frontiers in Human Neuroscience, 8, doi:10.3389/fnhum.2014.00068. Full:

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler’s identity as the “most beautiful theorem in mathematics”.[9] In another poll of readers that was conducted by Physics World in 2004, Euler’s identity tied with Maxwell’s equations (of electromagnetism) as the “greatest equation ever”

– Reid, Constance, From Zero to Infinity (Mathematical Association of America, various editions)

And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler’s formula and its applications in Fourier analysis.

Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills

Benjamin Peirce, a noted American 19th-century philosopher, mathematician, and professor at Harvard University, after proving Euler’s identity during a lecture, stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”.

– Maor, Eli, e: The Story of a number (Princeton University Press, 1998). ISBN 0-691-05854-7

It is said to be the equation that can link all of the constants of mathematics together.

The equation combines five of the most important numbers in mathematics.

These are:

1 – the basis of all other numbers

0 – the concept of nothingness

pi – the number that defines a circle

e – the number that underlies exponential growth i – the “imaginary” square root of -1

The numbers all have many practical applications, including communication, navigation, energy, manufacturing, finance, meteorology and medicine.

But that’s not all. Euler’s identity also contains the three most basic mathematical operations: addition, multiplication and exponentiation.

“Pi is an incredibly important number,” says Budd. “We have to calculate it to very high precision for modern technology such as GPS to work at all…  It can be used to describe the geometry of the world.”

“I tell my students that if this formula doesn’t completely blow them away then they simply have no soul,” says Chris Budd of the University of Bath in the UK.

It is roughly 3.14 but there are infinitely many more digits of approximate accuracy waiting for you to calculate.

Pi is an irrational/transcendental number.  -meaning the digits extending on forever -having no knowable final digit, no calculable limit.  -Without entering into a repeating pattern? without entering a fixed repetition pattern.

This number is the ratio of the circumference-length of any circle of any size to the length of the same circle’s largest centralized linear intersection/diameter.

??? TO WHAT EXTENT IS THE CIRCUMFERENCE OF A CIRCLE only approximate and not exact … it’s measured with arbitrary units …

– circle is an unvarying perfect curve –

??? how is the approach to exactness in the measurement of a curve/arc -the same or different- than the problem of squaring the circle …

Find and Read …

The DFT: An Owners’ Manual for the Discrete Fourier Transform

this is a rough draft of something

“I am that I am” is a primary archetypal phrase, by which the pages on this website are related.  It is a circular statement.

It is also linear, dualistic, and unifying.  It’s infinitely applicable to the cosmos, as is the circle geometry.  It’s an expression to contain all opposites.  A circle also contains all opposite qualities.

It could be said that “God” or “Existence” is an “Apeirogon” of infinite and infinitesimal possible Values in every dimension, representing all possible combinations and degrees of existence encompassed by both math and consciousness.

This isn’t hard mathematics or science, but a fun meditation for purposes of conceptualizing the mental and physical, potentiality and actuality.  It should be noted that the ancient Greeks’ contemplation of the circle was extensive and foundational to modern thought.

The practice and development of mathematics is based on making definitions within the infinite possibility of the formal imagination.

For me, one of the most interesting questions to ask a person is, “How many sides does a circle have?”  To which there are four possible answers.  (fourth answer, the inside or outside)  Zero is the most obvious answer that most people think of first.  Another two answers are one, one curved side; and infinity, an infinite number of sides.  All three answers are equally true.  A circle has no sides, one side, and an infinity of undetectably linear sides, of infinitesimally short length.  Or the inside and the outside.  –rewrite this paragraph in a more organized way–

Three other fundamental(not mathematical, but philosophical) questions, which I have fun asking, also have this set of //three answers.// (a similar set of answers)  They are, “How long does the present moment last?”  and “What numbers best represent the experience of God?  or “What is God? (from a quantitative perspective)” and “How many names does God have?”

Anyone will tell you God has no name, and they’re right.  Any name you give to god, means nothing, and is simply a placeholder for infinity.

Everyone will tell you that God has one name, and they’re neither right nor wrong.  People can refer to infinity with any word but will be coherent only to themselves.

And no one will tell you that God has an infinite number of names, but this is true.  Every word refers to God.

0, 1, infinity

The present moment can be seen to be so infinitesimally short in duration, that its transience is equivalent to emptiness.  The moment it’s here is the moment it’s gone.  Or it can be arbitrarily or relatively assigned a particular duration by a mind in time.  This is the One.  Any single quantification can be your choice.  It is indeed the function of the mind to assign a value to the present, whether it be mathematical or any other quality.  The third answer to this question of the present moment is everyone’s favorite..  that the present moment is eternal in duration; it cannot be fully contained in a finite mind.  This is intuitively true, and the most freeing of the answers.

God is fullness, emptiness, and the singularity.  These are true experiences in life.  The fullness of time, the emptiness of space, and the singular nature of mind in spacetime.  Focus on the One slows down time and makes it disappear.  Focus on the All speeds up time and makes it disappear.  Focus on Emptiness stops time where it is.  The genesis of stillness in living space is meditation.  Emptiness makes possible acceptance of the unity and oneness of reality, as well as its infinite possibility.

The essence of God is the union of (mind)infinity(boundless potentiality, the completed infinity of all possible forms) and (heart)emptiness(formless)(open acceptance, graceful receptivity) in the space of the present.

Some ancient Greeks indeed believed that zero, infinity, and one are not numbers at all.  The intuitive mind relates to them as transcendental qualities. //  starts with 2 ends with 10, and repeats from there..  but two can be viewed as a transcendental quality as well.. //  review chomsky video on math .. search chomsky math mathematics etc..

These three sacred quantities can be seen to be fundamentally related to the circle, and to the sphere and torus in higher dimensions.  At every point on the torus is a cross formed by the intersection of infinitely many circles.  Take a moment to ponder this at your leisure.

A Gateway to Modern Geometry, The Poincare Half-Plane

quote copy down Sine section to Regular Polygon section to clarify and extend visualization of Apeirogon in a logical way…   ^^^”squaring+the+circle”+apeirogon

—   MOVE TO PI AND INFINITY POST — delete from here

In the sense of physics it’s impossible to construct a perfect circle that is made of matter. But that’s only because physics imposes a strict sense of reality where you only believe something if you can measure it. To really “prove” that something is a circle, you would have to be able to measure its curvature in all places simultaneously up to infinite accuracy. This is not possible.

is squaring the circle a good general metaphor for approximation approaching accurate exactness in physical measurement?

how many ways are there of approximating pi?

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