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MATEMATICAS: RADIANES (FUERTE RELACION ESOTERICA CON EL GRIAL)
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EL EXAGONO Y LA RELACION CON EL PI=22/7-LAS MATEMATICAS CONFIRMAN LA RELACION EXAGONAL CON MARIA LA MAGDALENA EN EL MISMO DIA DE SU DIA EN LA IGLESIA CATOLICA, OSEA EL 22 DE JULIO (22/7)
 Temple of Man Schwaller de Lubicz page 177 Serpents in the Sky by John Anthony West
Noten que tenemos el SECTOR CIRCULAR, en un contexto matematico exagonal, la relacion matematica 22/11. Sabemos que la longitud de la circunferencia es igual al numero PI=3.14=22/7 por el DIAMETRO=RADIO*2 del circulo. En un contexto a que tenemos segun la figura superior la LONGITUD DE ARCO (PI/3-RADIANES) que es igual a 22 si hacemos las matematicas:
LONG. CIRC.=3.14*DIAMETRO=3.14*2*RADIO=22/7*2*21=22/7*2*7*3=(22*2*7*3)/7=22*6
LONG. CIRC.=22*6=132=33*4 (IMPRESIONANTE RELACION DEL NUMERO 33)
Noten que estamos haciendo referencia al exagono y es obvia su interrelacion con el planeta Mercurio, que insisto, esta interrelacionada con la ESTRELLA DE DAVID, OSEA EL HIJO DEL MISMO DAVID, QUE FUE EL REY SOL-O-MON. El 33, insisto, es una referencia a los dos triangulos, de la estrella de 6 puntas. Ahora, si la longitud del arco es igual a PI/3 (RADIANES) tenemos la relacion:
22/21=22/(7*3)=(22/7)/3=PI/3
IMPRESIONANTE DE QUE BELLAS SON LAS MATEMATICAS.
LA MISMA PALABRA REVOLU-SION ES UNA REFERENCIA AL NUMERO PI (LONGITUD DE LA CIRCUNFERENCIA ES IGUAL A PI X DIAMETRO.)
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LA FRECUENCIA DE LA CORRIENTE ALTERNA ESTA DISEÑADA EN FUNCION AL DIA DE MARIA MAGDALENA
EN FUNCION A QUE LA FRECUENCIA DE LA CORRIENTE O TENSION ALTERNA QUE ES DE 50 HERZ TENEMOS QUE LA VELOCIDAD ANGULAR U OMEGA, EN ESE MARCO ES DE 314 RADIANES POR SEGUNDO, OSEA EL MISMO NUMERO PI X 100=3.14 X100. SABEMOS QUE LA FRECUENCIA ES IGUAL A OMEGA (VELOCIDAD ANGULAR)/ (2*3.14). OSEA QUE OMEGA=2*3.14*50 (FRECUENCIA=50 HERZ)=100*3.14=314 RADIANES POR SEGUNDO.
4. Apocalipsis 22:13: Yo soy el Alfa y la OMEGA, el principio y el fin, el primero y el último
Movimiento circular
De Wikipedia, la enciclopedia libre
En cinemática, el movimiento circular (también llamado movimiento circunferencial) es el que se basa en un eje de giro y radio constante, por lo cual la trayectoria es una circunferencia. Si además, la velocidad de giro es constante (giro ondulatorio), se produce el movimiento circular uniforme, que es un caso particular de movimiento circular, con radio y centro fijos y velocidad angular constante.
En el movimiento circular hay que tener en cuenta algunos conceptos que serían básicos para la descripción cinemática y dinámica del mismo:
- Eje de giro: es la línea recta alrededor de la cual se realiza la rotación, este eje puede permanecer fijo o variar con el tiempo pero para cada instante concreto es el eje de la rotación (considerando en este caso una variación infinitesimal o diferencial de tiempo). El eje de giro define un punto llamado centro de giro de la trayectoria descrita (O).
- Arco: partiendo de un centro fijo o eje de giro fijo, es el espacio recorrido en la trayectoria circular o arco de radio unitario con el que se mide el desplazamiento angular. Su unidad es el radián (espacio recorrido dividido entre el radio de la trayectoria seguida, división de longitud entre longitud, adimensional por tanto).
- Velocidad angular: es la variación del desplazamiento angular por unidad de tiempo (omega minúscula,
 ).
- Aceleración angular: es la variación de la velocidad angular por unidad de tiempo (alfa minúscula,
 ).
En dinámica de los movimientos curvilíneos, circulares y/o giratorios se tienen en cuenta además las siguientes magnitudes:
- Momento angular (L): es la magnitud que en el movimiento rectilíneo equivale al momento lineal o cantidad de movimiento pero aplicada al movimiento curvilíneo, circular y/o giratorio (producto vectorial de la cantidad de movimiento por el vector posición, desde el centro de giro al punto donde se encuentra la masa puntual).
- Momento de inercia (I): es una cualidad de los cuerpos que depende de su forma y de la distribución de su masa y que resulta de multiplicar una porción concreta de la masa por la distancia que la separa al eje de giro.
- Momento de fuerza (M): o par motor es la fuerza aplicada por la distancia al eje de giro (es el equivalente a la fuerza agente del movimiento que cambia el estado de un movimiento rectilíneo).
Paralelismo entre el movimiento rectilíneo y el movimiento circular[editar]
A pesar de las diferencias evidentes en su trayectoria, hay ciertas similitudes entre el movimiento rectilíneo y el circular que deben mencionarse y que resaltan las similitudes y equivalencias de conceptos y un paralelismo en las magnitudes utilizadas para describirlos. Dado un eje de giro y la posición de una partícula puntual en movimiento circular o giratorio, para una variación de tiempo Δt o un instante dt, dado, se tiene:
Arco descrito o desplazamiento angular[editar]
Arco angular o desplazamiento angular es el arco de la circunferencia recorrido por la masa puntual en su trayectoria circular, medido en radianes y representado con la letras griegas  (phi) o  (theta). Este arco es el desplazamiento efectuado en el movimiento circular y se obtiene mediante la posición angular ( ó  ) en la que se encuentra en un momento determinado el móvil y al que se le asocia un ángulo determinado en radianes. Así el arco angular o desplazamiento angular se determinará por la variación de la posición angular entre dos momentos final e inicial concretos (dos posiciones distintas):
Siendo  ó  el arco angular o desplazamiento angular dado en radianes.
Si se le llama  al espacio recorrido a lo largo de la trayectoria curvilínea de la circunferencia de radio  se tiene que es el producto del radio de la trayectoria circular por la variación de la posición angular (desplazamiento angular):
En ocasiones se denomina  al espacio recorrido (del inglés "space"). Nótese que al multiplicar el radio por el ángulo en radianes, al ser estos últimos adimensionales (arco entre radio), el resultado es el espacio recorrido en unidades de longitud elegidas para expresar el radio.
Velocidad angular y velocidad tangencial[editar]
- Velocidad angular es la variación del arco angular o posición angular respecto al tiempo. Es representada con la letra
 (omega minúscula) y viene definida como:
Siendo la segunda ecuación la de la velocidad angular instantánea (derivada de la posición angular con respecto del tiempo).
- Velocidad tangencial de la partícula es la velocidad del objeto en un instante de tiempo (magnitud vectorial con módulo, dirección y sentido determinados en ese instante estudiado). Puede calcularse a partir de la velocidad angular. Si
 es el módulo la velocidad tangencial a lo largo de la trayectoria circular de radio R, se tiene que:
Aceleración angular y tangencial[editar]
La aceleración angular es la variación de la velocidad angular por unidad de tiempo y se representa con la letra:  y se la calcula:
Si at es la aceleración tangencial, a lo largo de la circunferencia de radio R, se tiene que:
Período y frecuencia[editar]
El período indica el tiempo que tarda un móvil en dar una vuelta a la circunferencia que recorre. Se define como:
La frecuencia es la inversa del periodo, es decir, las vueltas que da un móvil por unidad de tiempo. Se mide en hercios o s-1
Aceleración y fuerza centrípeta[editar]
Mecánica clásica[editar]
La aceleración centrípeta, también llamada normal o radial, afecta a un móvil siempre que éste realiza un movimiento circular, ya sea uniforme o acelerado. Se define como:
La fuerza centrípeta es la fuerza que produce en la partícula la aceleración centrípeta. Dada la masa del móvil, y basándose en la segunda ley de Newton ( ) se puede calcular la fuerza centrípeta a la que está sometido el móvil mediante la siguiente relación:
Mecánica relativista[editar]
En mecánica clásica la aceleración y la fuerza en un movimiento circular siempre son vectores paralelos, debido a la forma concreta que toma la segunda ley de Newton. Sin embargo, en relatividad especial la aceleración y la fuerza en un movimiento circular no son vectores paralelos a menos que se trate de un movimiento circular uniforme. Si el ángulo formado por la velocidad en un momento dado es  entonces el ángulo  formado por la fuerza y la aceleración es:
Para el movimiento rectilineo se tiene que  y por tanto  y para el movimiento circular uniforme se tiene  y por tanto también . En el resto de casos  . Para velocidades muy pequeñas y ángulos expresados en radianes se tiene:
Véase también[editar]
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Tao of the Tau
Tau is the Greek letter T or τ in lower case.
The symbol τ was used in mathematics to represent the golden ratio until the Greek letter Phi (Φ) gained prominance (after the first letter of Phidias, acclaimed sculptor of the Parthenon).
Tau is also known as the symbol representing the ratio of any circle’s circumference to its radius (equal to 2π).
Advocates for τ argue that radius is more fundamental to circles than diameter, and therefore, that τ (circumference divided by radius) is more fundamental than π (circumference divided by diameter). They think this makes formulas written in terms of τ express the mathematics more clearly than with π. -Source
Using τ instead of 2π makes a lot of sense to me:
- A circle is defined as all points in a plane a certain radius away from a center point.
- The unit circle has a radius of 1, not a diameter of 1.
- Angles measured in radians make more sense using τ rather than 2π because τ radians measures a full circle, so 1/4 circle is τ/4, 1/2 circle is τ/2 and so on.
However using τ is really a moot point because π already won the mindshare. However I find it interesting that Tau has been used to represent both the golden ratio (Φ) and the relationship of a circle to its radius (more commonly known as π).
I can think of one other symbol that encodes in its proportions both mathematical ideas: the Great Pyramid. Note that the proportions 4:π and Φ:1 are not taken at the same time but I’ve shown them together for simplicity, so read the left and right halves of the diagram separately.
“The Tau cross is also a symbol of the male or creative side of the deity, and is really a conventionalized form of the phallus.” -Freemasonry and the Ancient Gods by J.S. Ward
Crowley used an inverted Tau on the cover of his book Magick, which was “regarded in serious occult circles as a book of ultimate authority on sexual magic.” -Source
Seen right-side-up, the T is seen as the shape of the cross on which Jesus was crucified.
The TAU cross is preserved to modern Masonry under the symbol of the T square. This appears to be the oldest form of the cross extant.” -The Secret Teachings of All Ages by Manly Hall
There are several instances in the Bible of the Tau cross anticipating the Latin cross:
- The angel of death passed over the children of Israel who painted the Tau mark in goat’s or sheep’s blood above their doors to slay all Egyptian firstborn. This is better known as Passover. See Exodus 11:1 in the Brick Testament.
- The Tau mark was placed upon the foreheads of particular men to save them from being slain in yet another genocidal bloodbath depicted in Ezekiel 9 (unforunately this isn’t illustrated in Lego).
- The summit of Mt. Nebo in Jordan is the location of the Brazen Serpent Monument, featuring a snake wrapped around a Tau cross.
Image by Jerzy Strzelecki under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Mt. Nebo is where Moses saw the promised land of Israel (but wasn’t allowed to go there himself). You can see Jerusalem on a clear day from this monument.
Pope Benedict XVI visited the site on May 9, 2009, gave a speech, and looked out from the top of the mountain in the direction of Jerusalem. -Source
The fiery serpent comes from the Bible (Numbers 21:5-9):
- “And the people spake against God, and against Moses, Wherefore have ye brought us up out of Egypt to die in the wilderness? for there is no bread, neither is there any water; and our soul loatheth this light bread.
- And the LORD sent fiery serpents among the people, and they bit the people; and much people of Israel died.
- Therefore the people came to Moses, and said, We have sinned, for we have spoken against the LORD, and against thee; pray unto the LORD, that he take away the serpents from us. And Moses prayed for the people.
- And the LORD said unto Moses, Make thee a fiery serpent, and set it upon a pole: and it shall come to pass, that every one that is bitten, when he looketh upon it, shall live.
- And Moses made a serpent of brass, and put it upon a pole, and it came to pass, that if a serpent had bitten any man, when he beheld the serpent of brass, he lived.”
Check out this version illustrated with Lego. I like the Brick Testament because it consistently shows the character of this God. In this case God kills his chosen people with snakes because they complained about being hungry and thirsty. It’s easy to forget that and recall only that he healed those who didn’t complain with the Nehushtan as it is also known.
The 25th degree in Scottish Freemasonry is called is The Knight of the Brazen Serpent.
Here is a old masonic jewel that is probably from this degree.
Thanks to Rick for this image
The above image reminds one of the Rod of Asclepius. Here it is as the Star of Life, logo of the emergency medical services worldwide.
Also the emblem of the World Heath Organization underlines the fact that it is a specialized agency of the United Nations with its 33 sector grid laid over the world.
The symbol of a serpent climbing the pole is also a symbol of Kundalini, or life force rising up the spinal column.
These symbols all came from the Crux Ansata, better known as the Ankh. Think of it as a yoni on top of a lingam, or vesica piscis on top of a tau if you will. What better way to symbolize life then through sexual union? This has been pushed into the shadow in the last two thousand years in the West.
Thanks to reader Rick for pointing out that the Hindu Ardhanari is a composite androgynous form of the Hindu god Shiva and his consort Parvati: half male and half female, split down the middle. Note the Ankh is used here as a depiction of both male and female genitalia.
Image from Tantra: Cult of the Feminine by Andre van Lysebeth page 240
The Hindu androgyne reminds me of Baphomet whom the Templars revealed they worshipped under torture by the French Catholics in 1307.
Often mistaken for Satan, Baphomet represents the duality of male and female, as well as Heaven and Hell or night and day signified by the raising of one arm and the downward gesture of the other. -Source
I can’t help remember sex kitten Lady Gaga striking a similar pose in a Masonic Lodge (see Repeating Ones).
Image source
Look closely: Gaga’s left breast (on our right) is pushed up and her right breast is flattened much like Ardhanari. Gaga as Baphomet.
Image by Redtigerxyz under the Creative Commons Attribution 3.0 Unported license
Andre van Lysebeth shows in Tantra that the Om glyph actually represents the sexual act (female on the left and male on the right in this case) with what looks to me like the sun and moon in the upper right:
Image from Tantra: Cult of the Feminine by Andre van Lysebeth page 170
The Ankh is similar to the concept of the Tao in Chinese philosophy and religion. Opposites coming together form a dynamic balance.
Thank you Ian le Cheminant for emailing me this vitally important concept:
I had always associated synchronicity with Carl Jung but recently rereading Lama Anagarika Govinda’s The Inner Structure of the I Ching I was reminded that as a mode of thinking it is profoundly Chinese. Govinda writes: “All our reasoning is based on the law of cause and effect operating as a sequence. The Chinese do not reason so much along this horizontal line from past, through present to future; they reason perpendicularly, from what is in one place now to what is in another place now. In other words, they do not ask why, or from what past causes, a certain set of things is happening now; they ask, ‘What is the meaning of these things happening at this moment?’ The word Tao is the answer to this question. The present situation within and around oneself is Tao, for the present moment is life. Our memory of the past is contained in it as well as the potentiality for the future.”
Squaring the circle also comes to mind as a larger resolution of opposites, in the case of heaven and earth.
Earth Sun Moon by Scott Onstott
Pi brings together the linear and the curved, the limited and the unlimited, the human and the divine.
The Sword and the Ring by Scott Onstott
Slight dyslexia is all you need to get from Angels to Angles. In many areas of mathematics angles are measured in radians rather than degrees. One radian is equal to about 57 degrees (57.2957795…).
360 degrees = 2π or 6.283185… radians. An angle of 1 radian results in an arc with a length equal to the radius of the circle.
Tau is the 19th letter of the Greek alphabet. Three Taus therefore have a value of 3 x 19 = 57.
Triple Tau is an important Masonic symbol.
“This emblem, placed in the center of a Triangle and Circle – both emblems of Deity – constitutes the jewel of the Royal Arch as practiced in England, where it is so highly esteemed as to be called the “emblem of all emblems,” and “the grand emblem of Royal Arch Masonry.” -Source
Image by PeRshGo under the Creative Commons Attribution-Share Alike 3.0 Unported license.
The Royal Arch Degree Jew-el has a Star of David composed of two interlocking triangles (male and female) and a triple Tau at the bottom. This reminds me of the phrase triple bottom line or Hermes Trismegistus.
This ancient Alchemical emblem found in the ruins of Pompei suggests that the Star of David is a metaphor for the sexual act, a fact covered up by the SATOR magic square (see Paternoster Squares).
Robert Fowler discovered a cipher in Shakespeare’s sonnets dedication poem that requires one understand the Royal Arch Degree jewel in order to decode it. It’s a fascinating intellectual journey that I won’t belabour in this post.
However I will mention the decoded message refers to eternal life and a diagram that appears to depict Orion, an obelisk, a pyramid, the Star of David, and rays of the Sun. It is becoming harder not to see that Francis Bacon was involved in writing the works of “William Shakespeare.”
Listen to Robert Newman on The Shakespeare Project.
The 33rd degree jewel depicts four Taus forming a Greek cross.
“The quadruple tau, moreover, being composed entirely of ‘right angles, horizontals, and perpendiculars,’ contains within itself all the secret signs of Freemasonry, a fact which I am not permitted to further explain. It will, however, be apparent to every ‘bright mason,’ who can soon study them all out for himself.” -Stellar Theology and Masonic Astronomy by Robert Hewitt Brown
I’m not a mason but am bright enough to see at least on one level that the Quadruple Tau cross represents the great cross in the heavens as discussed in my Esoteric Astronomy video. The four beasts corresponding to the fixed signs in the aforementioned alchemical seal and Royal Arch degree jewel support this notion.
Here’s the Hindu version of the 33rd degree jewel: the Sri Yantra aka Sri Chakra. In addition to four taus, it features nine interlocking triangles, just as the 33rd degree jewel shows a nine pointed star and the brazen serpent jewel depicts a nine-pointed star.
Image by N.Manytchkine under the Creative Commons Attribution-Share Alike 3.0 Unported license
That brings me to a couple of very interesting buildings I’ve become aware of that I think encode the Quadruple Tau symbol. First up is Terrace on the Park at 52-11 111th Street in Flushing Meadows Park, Queens, NY. Do you see the solar 52 weeks in a year encoded with all the ones? The building was contructed as a helipad for the 1964 World’s Fair, which I plan to post about another time. One can say it has a special connection with the sky.
Thanks to John Handrinos who lives nearby the park for pointing out many of its secrets in plain sight.
Here is the official video about what goes on inside Terrace on the Park. Lots of people getting married inside Terrace on the Park seems quite appropriate symbolically.
Finally my thanks to Grant Chamberlin who discovered something very interesting about the CitiGroup Center (aka 601 Lexington Ave) in Manhattan.
The square base reminds me of the Quadruple Tau even though it is not highlighted as such.
Grant noticed that there are 32 full panels across the top of the T plus 2 half panels, making a total of 33 panels on each side. Just a coincidence or the Tao of the Tau?
©2012 SIPS Productions Inc. – All Rights Reserved.
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DÍAS DE CELEBRACIÓN DE π
Entre los que se encuentran el mismo 22 de Julio (22/7=3.1428), y curiosamente también el 21 de diciembre…
Pero el día oficial de π es el 3-14…
25/4 es el dia de SAN JUAN MARCOS, DIA NUMERO 227 DEL CALENDARIO COPTO
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Día Pi (π)
El día Pi es una fiesta no oficial que celebra la constante matemática pi (π). Se celebra el 14 de marzo en los países que siguen el formato mes / día que corresponden con los dígitos 3/14.
En los países que siguen el formato de fecha día / mes, la fiesta es después, el 22 de julio y se llama El día Pi por aproximación. Esto resulta de que el 22 de julio, o sea, el 22/7 es el valor aproximado de π (22 /7 = 3,14).
El día Pi el 14 de marzo también coincide con el cumpleaños de Albert Einstein. Otras fechas especiales del calendario del día Pi en 2015: Será la única vez del siglo que la celebración del sábado, será un Pi muy especial: Sólo se da una vez cada 100 años. Este año, no sólo el mes y el día de la fecha (3/14) corresponden a los dígitos de la constante matemática, los dígitos del año también lo hacen. Si se escribe la fecha en el formato mes / día / año, entonces los dígitos de la fecha 3/14/15 se corresponden con los 5 primeros dígitos de pi: 3.1415.
Pero eso no es todo. El día Pi de este año va un paso más allá. A las 9:26:53 am y las 09:26:53 pm, la fecha y la hora se corresponden exactamente a los 10 primeros dígitos de pi: 3,141592653. Muchas personas están llamando a este el segundo Pi. Aunque hay un cierto desacuerdo sobre si habrá dos o un segundo Pi. Los más "puristas" creen que sólo puede haber un segundo Pi el de las 9:26:53 am, considerando que a las 09:26:53 pm en el reloj de 24 horas son las 21:26:53.
A este respecto Jeffrey S. Rosenthal, profesor de Estadística de la Universidad de Toronto ha propuesto la designación del instante Pi: El instante exacto en el que la fecha y hora incluyen todos los dígitos de pi.
Cuenta atrás para el Segundo Pi
Una constante antigua y universal Pi (π), es una letra ampliamente conocida y una de las constantes matemáticas más reconocida, el diámetro en el espacio euclidiano o la relación del área de un círculo con su radio al cuadrado. El valor de PI es aproximadamente igual a 3.14159265, pero es un número irracional y su representación decimal nunca termina.
Día paléndromo
El día Pi fue fundado por Larry Shaw y se celebró por primera vez en 1988 en el Exploratorium de San Francisco.
El público y el personal del museo marcharon alrededor de un espacio circular y comiern pasteles de frutas.
Los estudiantes, los matemáticos y los entusiastas de las matemáticas de todo el mundo celebran el Día Pi participando en actividades y concursos alrededor de pi que incluye hacer y comer empanadas, convertir números tales como el tiempo o la edad de cada uno usando pi, y recitar los dígitos de pi o ver la famosa película en blanco y negro Pi (1988) de Darren Aronofsk
Días alternativos Pi y aproximación Pi se pueden celebrar en otras fechas del calendario como:
- 22 de julio: Cuando 22 se divide por 7, resulta igual a 3.14.
- 5 de Abril: Cuando han transcurrido 3,14 meses del año.
- 26 de Abril: La Tierra ha viajado dos radianes de su órbita en el día de hoy (25 de abril en los años bisiestos). Esta se celebra exactamente en el segundo 41 del minuto 23 de la cuarta hora, el 26 de abril o el día 116. (En los años bisiestos, se celebra exactamente en el tercer segundo del segundo minuto de la hora 12 el 25 de abril o el día 116 del año.)
- 10 de november: El día 314 del año (9 de noviembre en años bisiestos).
- 21 de Diciembre, a las 1:13 p.m.: El 355 día del año (20 de diciembre en los años bisiestos), que se celebra a las 1:13 de la aproximación china 355/113.
- http://elsoberadotecnologia.blogspot.com.ar/2015/03/dia-pi.html
22 DE JULIO=DIA DE MARIA MAGDALENA
25 DE ABRIL=DIA DE SAN MARCOS
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y medida en radianes 2π,
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y medida en radianes 2π,
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grados equivale un radián?
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aula.tareasplus.com
Regístrate para ver este video
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gabitos.com
Arco1.png
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en radianes (medida para
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grado medido a radianes,
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El área del triángulo sera 1/2
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Exprese en radianes los
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Tabla de Conversiones
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In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.
UPDATE!
 Triangle Formulas : Right TriangleC = A + B = Pi/2 radians = 90 degrees c^2 = a^2 + b^2 P = a + b + c s = (a+b+c)/2 K = ab/2
Particular case: Circle Inscribed in Triangle 3, 4, 5
Circle Inscribed in a Right Triangle
General case: Circle Inscribed in Triangle a, b, cWe can write, r = ab/(a + b + c) with c = sqrt(a^2 + b^2)
d = sqrt((a - 2r)^2+(b - 2r)^2)OR r = (a + b - c)/2 where c = Sqrt(a^2 + b^2)
Then (a - 2r)^2 + (b - 2r)^2 = (c - a)^2 + (c - b)^2 etc |
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Sinusoidal Waveforms
Generation of Sinusoidal Waveforms
In our tutorials about Electromagnetism, we saw how an electric current flowing through a conductor can be used to generate a magnetic field around itself, and also if a single wire conductor is moved or rotated within a stationary magnetic field, an “EMF”, (Electro-Motive Force) will be induced within the conductor due to this movement.
From this tutorial we learnt that a relationship exists between Electricity and Magnetism giving us, as Michael Faraday discovered the effect of “Electromagnetic Induction” and it is this basic principal that electrical machines and generators use to generate a Sinusoidal Waveform for our mains supply.
In the Electromagnetic Induction, tutorial we said that when a single wire conductor moves through a permanent magnetic field thereby cutting its lines of flux, an EMF is induced in it.
However, if the conductor moves in parallel with the magnetic field in the case of points A and B, no lines of flux are cut and no EMF is induced into the conductor, but if the conductor moves at right angles to the magnetic field as in the case of points C and D, the maximum amount of magnetic flux is cut producing the maximum amount of induced EMF.
Also, as the conductor cuts the magnetic field at different angles between points A and C, 0 and 90o the amount of induced EMF will lie somewhere between this zero and maximum value. Then the amount of emf induced within a conductor depends on the angle between the conductor and the magnetic flux as well as the strength of the magnetic field.
An AC generator uses the principal of Faraday’s electromagnetic induction to convert a mechanical energy such as rotation, into electrical energy, a Sinusoidal Waveform. A simple generator consists of a pair of permanent magnets producing a fixed magnetic field between a north and a south pole. Inside this magnetic field is a single rectangular loop of wire that can be rotated around a fixed axis allowing it to cut the magnetic flux at various angles as shown below.
Basic Single Coil AC Generator
As the coil rotates anticlockwise around the central axis which is perpendicular to the magnetic field, the wire loop cuts the lines of magnetic force set up between the north and south poles at different angles as the loop rotates. The amount of induced EMF in the loop at any instant of time is proportional to the angle of rotation of the wire loop.
As this wire loop rotates, electrons in the wire flow in one direction around the loop. Now when the wire loop has rotated past the 180o point and moves across the magnetic lines of force in the opposite direction, the electrons in the wire loop change and flow in the opposite direction. Then the direction of the electron movement determines the polarity of the induced voltage.
So we can see that when the loop or coil physically rotates one complete revolution, or 360o, one full sinusoidal waveform is produced with one cycle of the waveform being produced for each revolution of the coil. As the coil rotates within the magnetic field, the electrical connections are made to the coil by means of carbon brushes and slip-rings which are used to transfer the electrical current induced in the coil.
The amount of EMF induced into a coil cutting the magnetic lines of force is determined by the following three factors.
- • Speed – the speed at which the coil rotates inside the magnetic field.
- • Strength – the strength of the magnetic field.
- • Length – the length of the coil or conductor passing through the magnetic field.
We know that the frequency of a supply is the number of times a cycle appears in one second and that frequency is measured in Hertz. As one cycle of induced emf is produced each full revolution of the coil through a magnetic field comprising of a north and south pole as shown above, if the coil rotates at a constant speed a constant number of cycles will be produced per second giving a constant frequency. So by increasing the speed of rotation of the coil the frequency will also be increased. Therefore, frequency is proportional to the speed of rotation, ( ƒ ∝ Ν ) where Ν = r.p.m.
Also, our simple single coil generator above only has two poles, one north and one south pole, giving just one pair of poles. If we add more magnetic poles to the generator above so that it now has four poles in total, two north and two south, then for each revolution of the coil two cycles will be produced for the same rotational speed. Therefore, frequency is proportional to the number of pairs of magnetic poles, ( ƒ ∝ P ) of the generator where P = is the number of “pairs of poles”.
Then from these two facts we can say that the frequency output from an AC generator is:
Where: Ν is the speed of rotation in r.p.m. P is the number of “pairs of poles” and 60 converts it into seconds.
Instantaneous Voltage
The EMF induced in the coil at any instant of time depends upon the rate or speed at which the coil cuts the lines of magnetic flux between the poles and this is dependant upon the angle of rotation, Theta ( θ ) of the generating device. Because an AC waveform is constantly changing its value or amplitude, the waveform at any instant in time will have a different value from its next instant in time.
For example, the value at 1ms will be different to the value at 1.2ms and so on. These values are known generally as the Instantaneous Values, or Vi Then the instantaneous value of the waveform and also its direction will vary according to the position of the coil within the magnetic field as shown below.
Displacement of a Coil within a Magnetic Field
The instantaneous values of a sinusoidal waveform is given as the “Instantaneous value = Maximum value x sin θ ” and this is generalized by the formula.
Where, Vmax is the maximum voltage induced in the coil and θ = ωt, is the angle of coil rotation.
If we know the maximum or peak value of the waveform, by using the formula above the instantaneous values at various points along the waveform can be calculated. By plotting these values out onto graph paper, a sinusoidal waveform shape can be constructed.
In order to keep things simple we will plot the instantaneous values for the sinusoidal waveform at every 45o of rotation giving us 8 points to plot. Again, to keep it simple we will assume a maximum voltage, VMAX value of 100V. Plotting the instantaneous values at shorter intervals, for example at every 30o (12 points) or 10o (36 points) for example would result in a more accurate sinusoidal waveform construction.
Sinusoidal Waveform Construction
| Coil Angle ( θ ) |
0 |
45 |
90 |
135 |
180 |
225 |
270 |
315 |
360 |
| e = Vmax.sinθ |
0 |
70.71 |
100 |
70.71 |
0 |
-70.71 |
-100 |
-70.71 |
-0 |
The points on the sinusoidal waveform are obtained by projecting across from the various positions of rotation between 0o and 360o to the ordinate of the waveform that corresponds to the angle, θ and when the wire loop or coil rotates one complete revolution, or 360o, one full waveform is produced.
From the plot of the sinusoidal waveform we can see that when θ is equal to 0o, 180o or 360o, the generated EMF is zero as the coil cuts the minimum amount of lines of flux. But when θ is equal to 90o and 270o the generated EMF is at its maximum value as the maximum amount of flux is cut.
Therefore a sinusoidal waveform has a positive peak at 90o and a negative peak at 270o. Positions B, D, F and H generate a value of EMF corresponding to the formula e = Vmax.sinθ.
Then the waveform shape produced by our simple single loop generator is commonly referred to as a Sine Wave as it is said to be sinusoidal in its shape. This type of waveform is called a sine wave because it is based on the trigonometric sine function used in mathematics, ( x(t) = Amax.sinθ ).
When dealing with sine waves in the time domain and especially current related sine waves the unit of measurement used along the horizontal axis of the waveform can be either time, degrees or radians. In electrical engineering it is more common to use the Radian as the angular measurement of the angle along the horizontal axis rather than degrees. For example, ω = 100 rad/s, or 500 rad/s.
Radians
The Radian, (rad) is defined mathematically as a quadrant of a circle where the distance subtended on the circumference equals the radius (r) of the circle. Since the circumference of a circle is equal to 2π x radius, there must be 2π radians around a 360o circle, so 1 radian = 360o/2π = 57.3o. In electrical engineering the use of radians is very common so it is important to remember the following formula.
Definition of a Radian
Using radians as the unit of measurement for a sinusoidal waveform would give 2π radians for one full cycle of 360o. Then half a sinusoidal waveform must be equal to 1π radians or just π (pi). Then knowing that pi, π is equal to 3.142 or 22÷7, the relationship between degrees and radians for a sinusoidal waveform is given as.
Relationship between Degrees and Radians
Applying these two equations to various points along the waveform gives us.
The conversion between degrees and radians for the more common equivalents used in sinusoidal analysis are given in the following table.
Relationship between Degrees and Radians
| Degrees |
Radians |
Degrees |
Radians |
Degrees |
Radians |
| 0o |
0 |
135o |
3π
4
|
270o |
3π
2
|
| 30o |
π
6
|
150o |
5π
6
|
300o |
5π
3
|
| 45o |
π
4
|
180o |
π |
315o |
7π
4
|
| 60o |
π
3
|
210o |
7π
6
|
330o |
11π
6
|
| 90o |
π
2
|
225o |
5π
4
|
360o |
2π |
| 120o |
2π
3
|
240o |
4π
3
|
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|
The velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform. As the frequency of the waveform is given as ƒ Hz or cycles per second, the waveform has angular frequency, ω, (Greek letter omega), in radians per second. Then the angular velocity of a sinusoidal waveform is given as.
Angular Velocity of a Sinusoidal Waveform
and in the United Kingdom, the angular velocity or frequency of the mains supply is given as:
in the USA as their mains supply frequency is 60Hz it is therefore: 377 rad/s
So we now know that the velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform and which can also be called its angular velocity, ω. But we should by now also know that the time required to complete one revolution is equal to the periodic time, (T) of the sinusoidal waveform.
As frequency is inversely proportional to its time period, ƒ = 1/T we can therefore substitute the frequency quantity in the above equation for the equivalent periodic time quantity and substituting gives us.
The above equation states that for a smaller periodic time of the sinusoidal waveform, the greater must be the angular velocity of the waveform. Likewise in the equation above for the frequency quantity, the higher the frequency the higher the angular velocity.
Sinusoidal Waveform Example No1
A sinusoidal waveform is defined as: Vm = 169.8 sin(377t) volts. Calculate the RMS voltage of the waveform, its frequency and the instantaneous value of the voltage after a time of 6ms.
We know from above that the general expression given for a sinusoidal waveform is:
Then comparing this to our given expression for a sinusoidal waveform above of Vm = 169.8 sin(377t) will give us the peak voltage value of 169.8 volts for the waveform.
The waveforms RMS voltage is calculated as:
The angular velocity (ω) is given as 377 rad/s. Then 2πƒ = 377. So the frequency of the waveform is calculated as:
The instantaneous voltage Vi value after a time of 6mS is given as:
Note that the phase angle at time t = 6mS is given in radians. We could quite easily convert this to degrees if we wanted to and use this value instead to calculate the instantaneous voltage value. The angle in degrees will therefore be given as:
Sinusoidal Waveform
Then the generalised format used for analysing and calculating the various values of a Sinusoidal Waveform is as follows:
A Sinusoidal Waveform
In the next tutorial about Phase Difference we will look at the relationship between two sinusoidal waveforms that are of the same frequency but pass through the horizontal zero axis at different time intervals.
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Message 14 of 14 on the subject |
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Tao of the Tau
Tau is the Greek letter T or τ in lower case.
The symbol τ was used in mathematics to represent the golden ratio until the Greek letter Phi (Φ) gained prominance (after the first letter of Phidias, acclaimed sculptor of the Parthenon).
Tau is also known as the symbol representing the ratio of any circle’s circumference to its radius (equal to 2π).
Advocates for τ argue that radius is more fundamental to circles than diameter, and therefore, that τ (circumference divided by radius) is more fundamental than π (circumference divided by diameter). They think this makes formulas written in terms of τ express the mathematics more clearly than with π. –Source
Using τ instead of 2π makes a lot of sense to me:
- A circle is defined as all points in a plane a certain radius away from a center point.
- The unit circle has a radius of 1, not a diameter of 1.
- Angles measured in radians make more sense using τ rather than 2π because τ radians measures a full circle, so 1/4 circle is τ/4, 1/2 circle is τ/2 and so on.
However using τ is really a moot point because π already won the mindshare. However I find it interesting that Tau has been used to represent both the golden ratio (Φ) and the relationship of a circle to its radius (more commonly known as π).
I can think of one other symbol that encodes in its proportions both mathematical ideas: the Great Pyramid. Note that the proportions 4:π and Φ:1 are not taken at the same time but I’ve shown them together for simplicity, so read the left and right halves of the diagram separately.
“The Tau cross is also a symbol of the male or creative side of the deity, and is really a conventionalized form of the phallus.” –Freemasonry and the Ancient Gods by J.S. Ward
Crowley used an inverted Tau on the cover of his book Magick, which was “regarded in serious occult circles as a book of ultimate authority on sexual magic.” –Source
Seen right-side-up, the T is seen as the shape of the cross on which Jesus was crucified.
The TAU cross is preserved to modern Masonry under the symbol of the T square. This appears to be the oldest form of the cross extant.” –The Secret Teachings of All Ages by Manly Hall
There are several instances in the Bible of the Tau cross anticipating the Latin cross:
- The angel of death passed over the children of Israel who painted the Tau mark in goat’s or sheep’s blood above their doors to slay all Egyptian firstborn. This is better known as Passover. See Exodus 11:1 in the Brick Testament.
- The Tau mark was placed upon the foreheads of particular men to save them from being slain in yet another genocidal bloodbath depicted in Ezekiel 9 (unforunately this isn’t illustrated in Lego).
- The summit of Mt. Nebo in Jordan is the location of the Brazen Serpent Monument, featuring a snake wrapped around a Tau cross.
Image by Jerzy Strzelecki under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Mt. Nebo is where Moses saw the promised land of Israel (but wasn’t allowed to go there himself). You can see Jerusalem on a clear day from this monument.
Pope Benedict XVI visited the site on May 9, 2009, gave a speech, and looked out from the top of the mountain in the direction of Jerusalem. –Source
The fiery serpent comes from the Bible (Numbers 21:5-9):
- “And the people spake against God, and against Moses, Wherefore have ye brought us up out of Egypt to die in the wilderness? for there is no bread, neither is there any water; and our soul loatheth this light bread.
- And the LORD sent fiery serpents among the people, and they bit the people; and much people of Israel died.
- Therefore the people came to Moses, and said, We have sinned, for we have spoken against the LORD, and against thee; pray unto the LORD, that he take away the serpents from us. And Moses prayed for the people.
- And the LORD said unto Moses, Make thee a fiery serpent, and set it upon a pole: and it shall come to pass, that every one that is bitten, when he looketh upon it, shall live.
- And Moses made a serpent of brass, and put it upon a pole, and it came to pass, that if a serpent had bitten any man, when he beheld the serpent of brass, he lived.”
Check out this version illustrated with Lego. I like the Brick Testament because it consistently shows the character of this God. In this case God kills his chosen people with snakes because they complained about being hungry and thirsty. It’s easy to forget that and recall only that he healed those who didn’t complain with the Nehushtan as it is also known.
The 25th degree in Scottish Freemasonry is called is The Knight of the Brazen Serpent.
Here is a old masonic jewel that is probably from this degree.
Thanks to Rick for this image
The above image reminds one of the Rod of Asclepius. Here it is as the Star of Life, logo of the emergency medical services worldwide.
Also the emblem of the World Heath Organization underlines the fact that it is a specialized agency of the United Nations with its 33 sector grid laid over the world.
The symbol of a serpent climbing the pole is also a symbol of Kundalini, or life force rising up the spinal column.
These symbols all came from the Crux Ansata, better known as the Ankh. Think of it as a yoni on top of a lingam, or vesica piscis on top of a tau if you will. What better way to symbolize life then through sexual union? This has been pushed into the shadow in the last two thousand years in the West.
Thanks to reader Rick for pointing out that the Hindu Ardhanari is a composite androgynous form of the Hindu god Shiva and his consort Parvati: half male and half female, split down the middle. Note the Ankh is used here as a depiction of both male and female genitalia.
Image from Tantra: Cult of the Feminine by Andre van Lysebeth page 240
The Hindu androgyne reminds me of Baphomet whom the Templars revealed they worshipped under torture by the French Catholics in 1307.
Often mistaken for Satan, Baphomet represents the duality of male and female, as well as Heaven and Hell or night and day signified by the raising of one arm and the downward gesture of the other. –Source
I can’t help remember sex kitten Lady Gaga striking a similar pose in a Masonic Lodge (see Repeating Ones).
Image source
Look closely: Gaga’s left breast (on our right) is pushed up and her right breast is flattened much like Ardhanari. Gaga as Baphomet.
Image by Redtigerxyz under the Creative Commons Attribution 3.0 Unported license
Andre van Lysebeth shows in Tantra that the Om glyph actually represents the sexual act (female on the left and male on the right in this case) with what looks to me like the sun and moon in the upper right:
Image from Tantra: Cult of the Feminine by Andre van Lysebeth page 170
The Ankh is similar to the concept of the Tao in Chinese philosophy and religion. Opposites coming together form a dynamic balance.
Thank you Ian le Cheminant for emailing me this vitally important concept:
I had always associated synchronicity with Carl Jung but recently rereading Lama Anagarika Govinda’s The Inner Structure of the I Ching I was reminded that as a mode of thinking it is profoundly Chinese. Govinda writes: “All our reasoning is based on the law of cause and effect operating as a sequence. The Chinese do not reason so much along this horizontal line from past, through present to future; they reason perpendicularly, from what is in one place now to what is in another place now. In other words, they do not ask why, or from what past causes, a certain set of things is happening now; they ask, ‘What is the meaning of these things happening at this moment?’ The word Tao is the answer to this question. The present situation within and around oneself is Tao, for the present moment is life. Our memory of the past is contained in it as well as the potentiality for the future.”
Squaring the circle also comes to mind as a larger resolution of opposites, in the case of heaven and earth.
Earth Sun Moon by Scott Onstott
Pi brings together the linear and the curved, the limited and the unlimited, the human and the divine.
The Sword and the Ring by Scott Onstott
Slight dyslexia is all you need to get from Angels to Angles. In many areas of mathematics angles are measured in radians rather than degrees. One radian is equal to about 57 degrees (57.2957795…).
360 degrees = 2π or 6.283185… radians. An angle of 1 radian results in an arc with a length equal to the radius of the circle.
Tau is the 19th letter of the Greek alphabet. Three Taus therefore have a value of 3 x 19 = 57.
Triple Tau is an important Masonic symbol.
“This emblem, placed in the center of a Triangle and Circle – both emblems of Deity – constitutes the jewel of the Royal Arch as practiced in England, where it is so highly esteemed as to be called the “emblem of all emblems,” and “the grand emblem of Royal Arch Masonry.” –Source
Image by PeRshGo under the Creative Commons Attribution-Share Alike 3.0 Unported license.
The Royal Arch Degree Jew-el has a Star of David composed of two interlocking triangles (male and female) and a triple Tau at the bottom. This reminds me of the phrase triple bottom line or Hermes Trismegistus.
This ancient Alchemical emblem found in the ruins of Pompei suggests that the Star of David is a metaphor for the sexual act, a fact covered up by the SATOR magic square (see Paternoster Squares).
Robert Fowler discovered a cipher in Shakespeare’s sonnets dedication poem that requires one understand the Royal Arch Degree jewel in order to decode it. It’s a fascinating intellectual journey that I won’t belabour in this post.
However I will mention the decoded message refers to eternal life and a diagram that appears to depict Orion, an obelisk, a pyramid, the Star of David, and rays of the Sun. It is becoming harder not to see that Francis Bacon was involved in writing the works of “William Shakespeare.”
Listen to Robert Newman on The Shakespeare Project.
The 33rd degree jewel depicts four Taus forming a Greek cross.
“The quadruple tau, moreover, being composed entirely of ‘right angles, horizontals, and perpendiculars,’ contains within itself all the secret signs of Freemasonry, a fact which I am not permitted to further explain. It will, however, be apparent to every ‘bright mason,’ who can soon study them all out for himself.” –Stellar Theology and Masonic Astronomy by Robert Hewitt Brown
I’m not a mason but am bright enough to see at least on one level that the Quadruple Tau cross represents the great cross in the heavens as discussed in my Esoteric Astronomy video. The four beasts corresponding to the fixed signs in the aforementioned alchemical seal and Royal Arch degree jewel support this notion.
Here’s the Hindu version of the 33rd degree jewel: the Sri Yantra aka Sri Chakra. In addition to four taus, it features nine interlocking triangles, just as the 33rd degree jewel shows a nine pointed star and the brazen serpent jewel depicts a nine-pointed star.
Image by N.Manytchkine under the Creative Commons Attribution-Share Alike 3.0 Unported license
That brings me to a couple of very interesting buildings I’ve become aware of that I think encode the Quadruple Tau symbol. First up is Terrace on the Park at 52-11 111th Street in Flushing Meadows Park, Queens, NY. Do you see the solar 52 weeks in a year encoded with all the ones? The building was contructed as a helipad for the 1964 World’s Fair, which I plan to post about another time. One can say it has a special connection with the sky.
Thanks to John Handrinos who lives nearby the park for pointing out many of its secrets in plain sight.
Here is the official video about what goes on inside Terrace on the Park. Lots of people getting married inside Terrace on the Park seems quite appropriate symbolically.
http://www.secretsinplainsight.com/2012/01/16/tao-of-the-tau/
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The definition of 1 radian. The red line (radius) and blue line (arc length) are the same length. By Stannered, CC-SA-BY-3.0 |
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Radians are another way to measure angles.
There are 2*pi radians in a full turn so 360 degrees = 2*pi radians
180 degress = pi radians
90 degrees = pi/2 radians etc
Quite often we talk about measures of radians in terms of multiples (or fractions) of pi.
To convert from degrees to radians you divide by 180 and multiply by pi.
To convert from radians to degrees you do the reverse and divide by pi and multiply by 360.
The term "two radians" therefore is the size of an angle. We can think of it in degrees if that makes more sense by converting it as described above.
2/pi*180 = about 114.6 degrees.
If you are doing any calculus you should always make sure you use radians for your angles.
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File:Circle radians.gif
From Wikipedia, the free encyclopedia
No higher resolution available.
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| 0 degrees |
0 radians |
| 30 degrees |
pi/6 radians |
| 45 degrees |
pi/4 radians |
| 60 degrees |
pi/3 radians |
| 90 degrees |
pi/2 radians |
| 120 degrees |
2pi/3 radians |
| 135 degrees |
3pi/4 radians |
| 150 degrees |
5pi/6 radians |
| 180 degrees |
pi radians |
| 210 degrees |
7pi/6 radians |
| 225 degrees |
5pi/4 radians |
| 240 degrees |
4pi/3 radians |
| 270 degrees |
3pi/2 radians |
| 300 degrees |
5pi/3 radians |
| 315 degrees |
7pi/4 radians |
| 330 degrees |
11pi/6 radians |
| 360 degrees |
2pi radians |
|
|
|
|
|
| 0 degrees |
0 radians |
| 30 degrees |
pi/6 radians |
| 45 degrees |
pi/4 radians |
| 60 degrees |
pi/3 radians |
| 90 degrees |
pi/2 radians |
| 120 degrees |
2pi/3 radians |
| 135 degrees |
3pi/4 radians |
| 150 degrees |
5pi/6 radians |
| 180 degrees |
pi radians |
| 210 degrees |
7pi/6 radians |
| 225 degrees |
5pi/4 radians |
| 240 degrees |
4pi/3 radians |
| 270 degrees |
3pi/2 radians |
| 300 degrees |
5pi/3 radians |
| 315 degrees |
7pi/4 radians |
| 330 degrees |
11pi/6 radians |
| 360 degrees |
2pi radians |
NOTEN LA SIMILITUD CON EL 9/11, EN EL MARCO AL 33 Y AL NUMERO PI (9=6 DADO VUELTA) |
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Image from The Contiunuum Encyclopedia of Symbols by Udo Becker
