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Show how a unit circle with a radius of 1 and angles x in radians can be used to graph the sine and cosine functions and to prove that cos²(x) + sin²(x) = 1 using the Pythagorean theorem.
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"A fool is known by his speech, and a wise man by silence."
Pythagoras, a Greek mathematician and philosopher, lived around 570-495 BCE. While much of his life is shrouded in mystery and legend, here are five lesser-known or potentially rare facts about Pythagoras:
Secretive Cult-Like Community: Pythagoras founded a secretive and exclusive community in Croton, Italy, known as the Pythagorean Brotherhood. The members of this community were bound by strict rules, including a vow of silence and a communal way of life. The group was not only interested in mathematics but also philosophy, mysticism, and religious practices.
Numerical Mysticism: Pythagoras and his followers believed in the mystical significance of numbers. They thought that numbers held inherent qualities and mystical properties. For example, they associated odd numbers with masculinity and even numbers with femininity. The number 10 was particularly significant and seen as a perfect and divine number.
Pythagorean Theorem was known before Pythagoras: While Pythagoras is credited with the famous Pythagorean Theorem, which relates the lengths of the sides of a right-angled triangle, evidence suggests that the theorem was known to the Babylonians and Indians before Pythagoras. Pythagoras, however, is often credited with its formal proof and popularization in ancient Greece.
Vegetarianism and Taboos: Pythagoras and his followers were known for their strict dietary restrictions. They practiced vegetarianism, believing that it was essential for purifying the soul. They also had various food-related taboos, such as avoiding beans, possibly due to their belief in the impurity associated with certain legumes.
Pythagorean Cup: Pythagoras is associated with the invention of the Pythagorean cup, also known as a "greedy cup" or "tantalus cup." It's a clever drinking vessel with a hidden mechanism that causes the cup to drain its contents if filled beyond a certain level. This invention is often attributed to Pythagoras as a symbolic lesson in moderation and self-control.
Pythagoras had a phobia of beans. It was his fear of beans that caused his death. When attackers chased him into a field of beans, he refused to enter and was killed instead. #FACT
DSEQEC — 2
2018, Reginald BrooksÂ
more…Â
DSEQEC — 1
2018, Reginald BrooksÂ
more…Â
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rbrooksdesign:
DSEQEC-2
DSEQEC: Double-Slit Experiment — Quantum Entanglement Conjecture Reginald Brooks
On the BIM (BBS-ISL Matrix), ALL Pythagorean Triples — parent, Primitive Pythagorean Triples (PPTs) and child, non-Primitive PTs (nPPTs) — are found.
Because the BIM extends all Inverse Square Law (ISL) information to infinity, and, knowing any one matrix grid cell value and location, one (Nature) ubiquitously knows ALL the BIM values at ALL grid cell locations, and because the PTs, as part of the BIM, are also ubiquitously known, NO FASTER THAN LIGHT signal is required to extend the information to its ENTANGLED cohort particle regardless of the degree of space or time separation, i.e. SpaceTime (ST) separation.
Entangled ST unit particles, like the photon, coherently share the same quantum state, only with equal but opposite values. The Conservation Laws (energy, charge, spin angular momentum) require the combined coherent state of the entangled ST unit particles to be the net value of its individual units when combined. Spin is often the quantum state value measured to label two equal but opposite values contributed. Spin may be designated as up-down, left-right, + or -, …
It is proposed in the DSEQEC that the PTs represent the quantum state values of spin, with the direction of travel of the spin vector that is measured being one of the sides of the PT.
Since the BIM itself is bilaterally symmetrical, and, because every PT on the BIM is represented on a Row (or Column), and can be depicted in either “left-” or “right-“handed iterations, there are 4 total iterations of any given PT. Â
Ones particular axial frame of reference when measuring the spin vector as up-down, left-right, + or -,…. ALWAYS results in the measurement reading of the opposite, but coherently entangled particle spin vector to be INSTANTLY equal but opposite precisely because the spin information is ALWAYS known. The movement in opposite direction ensures that, given the same axial frame of reference, the opposite will always be chosen.
One may argue that the measurement is collapsing the wave-probability function in which both spin vector directions simultaneously exist and it is the act of measuring that collapses the wave function to select one or the other spin vector direction. And in one sense, that is true. But NOT in the sense that some hidden information traveling faster than light is then communicating, via a signal, to the other particle to register the opposite spin vector direction.
NO. There are is NO hidden variable information and there is NO faster than light signal pushing information to the other entangled particle.
What is happening is that the choice of measurement to utilize a specific and consistent axial frame of reference to register the spin vector angular momentum direction crucially pre-selects the COMPLETELY, SIMULTANEOUSLY KNOWN INFORMATION of the other entangled ST unit particle, giving the equal, but opposite spin vector direction when measured with the same axial frame of reference.
The DSEQEC is consistent with and provides a complementary theoretical basis for the ER = EPR work of Leonard Susskind, Juan Maldacena and Mark Van Raamsdonk. The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.
But what of the Double-Slit Experiment? The same KNOWN INFORMATION of the PTs on the BIM means that even a single ST unit particle — be it a photon or an electron — can interfere with itself in going through the two slits because the INFORMATION of BOTH possible paths is KNOWN from the start. Now, it is the process of detecting (measuring) which slit the ST particle went through that destroys the heretofor entangled INFORMATION, resulting in the loss of the interference pattern.
A tip of the hat to Jacob Kafka’s “Rough Animator” app.
more… see DSEQEC
Click HERE for the TPISC — MathspeedST — BBS-ISL Matrix Media Center
including iBooks (free for Teachers, Educators and Students)
rbrooksdesign:
DSEQEC-1
DSEQEC: Double-Slit Experiment — Quantum Entanglement Conjecture Reginald Brooks
 On the BIM (BBS-ISL Matrix), ALL Pythagorean Triples — parent, Primitive Pythagorean Triples (PPTs) and child, non-Primitive PTs (nPPTs) — are found.
 Because the BIM extends all Inverse Square Law (ISL) information to infinity, and, knowing any one matrix grid cell value and location, one (Nature) ubiquitously knows ALL the BIM values at ALL grid cell locations, and because the PTs, as part of the BIM, are also ubiquitously known, NO FASTER THAN LIGHT signal is required to extend the information to its ENTANGLED cohort particle regardless of the degree of space or time separation, i.e. SpaceTime (ST) separation.Â
Entangled ST unit particles, like the photon, coherently share the same quantum state, only with equal but opposite values. The Conservation Laws (energy, charge, spin angular momentum) require the combined coherent state of the entangled ST unit particles to be the net value of its individual units when combined. Spin is often the quantum state value measured to label two equal but opposite values contributed. Spin may be designated as up-down, left-right, + or -, …Â
 It is proposed in the DSEQEC that the PTs represent the quantum state values of spin, with the direction of travel of the spin vector that is measured being one of the sides of the PT.
  Since the BIM itself is bilaterally symmetrical, and, because every PT on the BIM is represented on a Row (or Column), and can be depicted in either “left-” or “right-“handed iterations, there are 4 total iterations of any given PT. Â
 Ones particular axial frame of reference when measuring the spin vector as up-down, left-right, + or -,…. ALWAYS results in the measurement reading of the opposite, but coherently entangled particle spin vector to be INSTANTLY equal but opposite precisely because the spin information is ALWAYS known. The movement in opposite direction ensures that, given the same axial frame of reference, the opposite will always be chosen.
  One may argue that the measurement is collapsing the wave-probability function in which both spin vector directions simultaneously exist and it is the act of measuring that collapses the wave function to select one or the other spin vector direction. And in one sense, that is true. But NOT in the sense that some hidden information traveling faster than light is then communicating, via a signal, to the other particle to register the opposite spin vector direction.Â
  NO. There are is NO hidden variable information and there is NO faster than light signal pushing information to the other entangled particle.Â
What is happening is that the choice of measurement to utilize a specific and consistent axial frame of reference to register the spin vector angular momentum direction crucially pre-selects the COMPLETELY, SIMULTANEOUSLY KNOWN INFORMATION of the other entangled ST unit particle, giving the equal, but opposite spin vector direction when measured with the same axial frame of reference.Â
  The DSEQEC is consistent with and provides a complementary theoretical basis for the ER = EPR work of Leonard Susskind, Juan Maldacena and Mark Van Raamsdonk. The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.Â
 But what of the Double-Slit Experiment? The same KNOWN INFORMATION of the PTs on the BIM means that even a single ST unit particle — be it a photon or an electron — can interfere with itself in going through the two slits because the INFORMATION of BOTH possible paths is KNOWN from the start. Now, it is the process of detecting (measuring) which slit the ST particle went through that destroys the heretofor entangled INFORMATION, resulting in the loss of the interference pattern.
  A tip of the hat to Jacob Kafka’s “Rough Animator” app.
more… see DSEQECÂ
Click HERE for the TPISC — MathspeedST — BBS-ISL Matrix Media Center
including iBooks (free for Teachers, Educators and Students)
The Pythagorean Theorem Is Not What You Think
20 Apr 2026 #PythagoreanTheorem #MathExplained #Geometry There’s a reason the Pythagorean theorem keeps appearing in places it shouldn’t. From coordinate geometry to signal processing… from circles to spacetime… It’s not just a formula about triangles. It’s a description of something much deeper. In this video: • Why a² + b² = c² is really about areas, not lengths • Proofs from Bhaskara, Euclid,…
