After being (very unexpectedly, may I add) on Tumblr’s radar and receiving many new followers, I feel as popular as Isaac Newton, if only I hadhis brain! We’ve all heard of him but do we all know what is he actually known for? Laws!
Yes you heard me, laws. More specifically, Newton’s laws of motion.
Picture above: Isaac Newton (1643-1727), the man that changed our understanding of the Universe with his three laws of motion.
Before learning these laws, we might want to know what a force actually is and what it does!
A force can change the speed of an object and it can also change the form of the object. The unit for force is (big suprise) N, for Newton. We can say a force is any influence that causes an object to change, for example it’s movement, direction or even geometric construction could change.
Picture above: Forces can also be described as a push or a pull on an object. For example gravity or magnetism.
Another thing that can be handy to know, is what a vector quantity is. As written in one of my past posts about time: Scalars are quantities that are fully described by a magnitude alone (Time and Mass), whilst Vectors are quantities that are fully described by both a magnitude and a direction (Velocity and Force).
The first law:
If the net force, which means the overall force acting on an object, is zero, then the velocity of the object is constant.
If an object is stood still, then the net force will equal zero. Why? Imagine a table placed on the ground, there will be the gravity force keeping the table on the ground and a force coming from the ground up against the table (called the Normal Force: http://en.wikipedia.org/wiki/Normal_force). These two forces will be of the same size, in opposite directions.
Picture above: Fn = Normal Force
As the object is stood still, Newton’s first law says the net force is equal to zero.
This law states that the net force of an object is equal to the mass of the object and the acceleration. When an object is influenced by forces, then the object has an acceleration in the same direction as the sum of the forces (the net force).
When two bodies exert forces on each other, the forces will be equally big and in opposite directions. So if a first body exerts a force F1 on a second body, then the second body will at the same time exert a force on the first body. F1 and F2 are then equal in magnitude and opposite in direction.
So next time you give someone a hug or… even punch them if you feel that angry, just remember Newton’s third law!
The four Cartesian quadrants provide the two-dimensional analogue of the number line and its graphic representation in Cartesian coordinate space. This is the true native habitat of the square and, by implication, of square root. Because Enlightenment mathematicians found fit to define square root in a different context inadvertently -that of the number line- we will find it necessary to devise a different name for what ought rightly to have been called square root, but wasn’t. I propose that we retain the existent definition of tradition and refer to the new relationship between opposite numbers in the square, that is to say, opposite vertices through two dimensions or antipodal numbers, as contra-square root.[1]
Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number[2]ofany dimension multiplied by itself gives as result the identity element of that express dimension. For the native two-dimensional context of the square the identity element is OLD YANG, the bigram composed of two stacked yang (+) Lines, which corresponds to yang (+1), the identity element in the one-dimensional context of the number line. In a three-dimensional context, the identity element is the trigram HEAVEN which is composed of three stacked yang (+) Lines. The crucial idea here is that the identity element differs for each dimensional context, and whatever that context might be, it produces no change when in the operation of multiplication it acts as operator on any operand within the stated dimension.[3]
As a corollary it can be stated that any number in any dimension n composed of any combination of yang Lines (+1) and yin Lines (-1) if multiplied by itself (i.e., squared) produces the identity element for that dimension. In concrete terms this means, for example, that any bigram multiplied by itself equals the bigram OLD YANG; any of eight trigrams multiplied by itself equals the trigram HEAVEN; and any of the sixty-four hexagrams multiplied by itself equals the hexagram HEAVEN; etc. (valid for any and all dimensions without exception). Consequently, the number of roots the identity element has in any dimension n is equal to the number 2n, these all being real roots in that particular dimension.
Similar contextual analysis would show that the inversion element of any dimension n has 2n roots of the kind we have agreed to refer to as contra-square roots in deference to the Mathematics Establishment.[4]
That leads us to the possibly startling conclusion that in every dimension n there is an inversion element that has the same number of roots as the identity elementandall of them are real roots. For two dimensions the two pairs that satisfy the requirement are bigram pairs
For one dimension there is only a single pair that satisfies. That is (surprise, surprise) yin(-1)/yang (+1). What it comes down to is this:
If we are going to continue to insist on referring to square root in terms of the one-dimensional number line, then
+1 has two real roots of the traditional variety, +1 and -1
-1 has two real roots of the newly defined contra variety, +1/-1 and -1/+1
So where do imaginary numbers and quaternions fit in all this? The short answer is they don’t. Imaginary numbers entered the annals of human thought through error. There was a pivotal moment[5] in the history of mathematics and science, an opportunity to see that there are in every dimension two different kinds of roots - - - what has been called square root and what we are calling contra-square roots. Enlightenment mathematicians and philosophers essentially allowed the opportunity to slip through their fingers unnoticed.[6]
Descartes at least saw through the veil. He called the whole matter of imaginary numbers ‘preposterous’. It seems his venerable opinion was overruled though. Isaac Newton had his say in the matter too. He claimed that roots of imaginary numbers “had to occur in pairs.” And yet another great mathematician, philosopher opined. Gottfried Wilhelm Leibniz, in 1702 characterized √−1 as “that amphibian between being and non-being which we call the imaginary root of negative unity.” Had he but preserved such augury conspicuously in mind he might have elaborated the concept of probable numbers in the 18th century. If only he had truly understood the I Ching, instead of dismissing it as a primitive articulation of his own binary number system.
Image: The four quadrants of the Cartesian plane. By convention the quadrants are numbered in a counterclockwise direction. It is as though two number lines were placed together, one going left-right, and the other going up-down to provide context for the two-dimensional plane. Sourced from Math Is Fun.
Notes
[1] My preference might be for square root to be redefined from the bottom up, but I don’t see that happening in our lifetimes. Then too this way could be better.
[2] By the term unit number, I intend any number of a given dimension that consists entirely of variant elements of the number one (1) in either its positive or negative manifestation. Stated differently, these are vectors having various different directions within the dimension, but all of scalar value -1 (yin) or +1 (yang). All emblems of I Ching symbolic logic satisfy this requirement. These include the Line, bigram, trigram, tetragram, and hexagram. In any dimension n there exist 2n such emblems. In sum, for our purposes here, a unit number is any of the set of numbers, within any dimension n, which when self-multiplied (squared) produces the multiplicative identity of that dimension which is itself, of course, a member of the set.
ADDENDUM (01 MAY 2016): I’ve since learned that mathematics has a much simpler way of describing this. It calls all these unit vectors. Simple, yes?
[3] I think it fair to presume that this might well have physical correlates in terms of quantum mechanical states or numbers. Here’s a thought: why would it be necessary that all subatomic particles exist in the same dimension at all times given that they have a playing field of multiple dimensions, - some of them near certainly beyond the three with which we are familiar? And why would it not be possible for two different particles to be stable and unchanging in their different dimensions, yet become reactive and interact with one another when both enter the same dimension or same amplitude of dimension?
[4] Since in any contra-pair (antipodal opposites) of any dimension, either member of the pair must be regarded once as operator and once as operand. So for the two-dimensional square, for example, there are two antipodal pairs (diagonals) and either vertex of each can be either operator or operand. So in this case, 2 x 2 = 4. For trigrams there are four antipodal pairs, and 2 x 4 = 8. For hexagrams there are thirty-two antipodal pairs and 2 x 32 = 64. In general, for any dimension n there are 2 x 2n/2 = 2n antipodal pairs or contra-roots.
[5] Actually lasting several centuries, from about the 16th to the 19th century. Long enough, assuredly, for the error to have been discovered and corrected. Instead, the 20th century dawned with error still in place, and physicists eager to explain the newly discovered bewildering quantum phenomena compounded the error by latching onto √−1 and quaternions to assuage their confusion and discomfiture. This probably took place in the early days of quantum mechanics when the Bohr model of the atom still featured electrons as traveling in circular orbits around the nucleus or soon thereafter, visions of minuscule solar systems still fresh in the mind. At that time rotations detailed by imaginary numbers and quaternions may have still made some sense. Such are the vagaries of history.
[6] I think an important point to consider is that imaginary and complex numbers were, -to mathematicians and physicists alike,- new toys of a sort that enabled them to accomplish certain things they could not otherwise. They were basically tools of empowerment which allowed manipulation of numbers and points on a graph more easily or conveniently. They provided their controllers a longed for power over symbols, if not over the real world itself. In the modern world ever more of what we humans do and want to do involves manipulation of symbols. Herein, I think, lies the rationale for our continued fascination with and dependence on these tools of the trade. They don’t need to actually apply to the world of nature, the noumenal world, so long as they satisfy human desire for domination over the world of symbols it has created for itself and in which it increasingly dwells, to a considerable degree apart from the natural world’s sometimes seemingly too harsh laws.
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
For my 20th Portraits at the Pub we had another PATP model that like Rick we had just before the first Lockdown – Jules! As before I started with ink drawings in the A4 sketchbook, including a post that was the mirror of last time but drawn far better! I was using the Evergood french flex vintage pen, a recent purchase and refurb – only £17 on eBay, fixed by myself with a new ink sac fitted. It’s…
A Very Short Fact: On this day in 1643, English mathematician, physicist, astronomer, theologian, and philosopher Sir Isaac Newton was born.
“According to the calendar then in use in England, Newton was born on Christmas Day 1642 (4 January 1643 in most of Continental Europe). The first decade of his life witnessed the horror of the civil wars between parliamentary and royalist forces in the 1640s, culminating in the beheading of Charles I in January 1649. His uncle and stepfather were rectors of local parishes, and they seem to have existed without much harassment from the church authorities convened by Parliament to check for religious ‘abuses’. In his second decade he lived under the radical Protestant Commonwealth, which was replaced in 1660 when Charles II was restored to the throne. Newton was born into a relatively prosperous family and was brought up in a devout atmosphere. His father, also Isaac, was a yeoman farmer who in December 1639 inherited both land and a handsome manor in the Lincolnshire parish of Woolsthorpe. His mother, Hannah Ayscough, came from the lower gentry and (as was common for the period) seems to have been educated at only a rudimentary level. Nevertheless, her brother William had graduated from Trinity College Cambridge in the 1630s and would be influential in directing Newton to the same institution.
Newton’s father, apparently unable to sign his name, died in early October 1642, almost three months before the birth of his son. Newton told Conduitt that he had been a tiny and sick baby, thought to be unlikely to survive; two women sent to get help from a local gentlewoman stopped to sit down on the way there, as they were certain the baby would be dead on their return. Surviving against the odds, Newton was brought up by his mother until the age of 3, when she was approached with an offer of marriage by Barnabas Smith, an ageing vicar of a local parish. Smith was wealthy, and they married in January 1646 after he had promised to leave some land to her first born. Spending most of her time with her new spouse, she produced three more children before his death in 1653 (one of whom would be the mother of Catherine Conduitt). Although John Conduitt waxed lyrical about Hannah’s general virtues, and was careful to point out that she was ‘an indulgent parent’ to all the children, he emphasized that young Isaac was her favourite. Whatever the truth of this, Newton’s own evidence indicates that, as a teenager, he had an extremely difficult relationship with his mother, and historians have always found it difficult to make Conduitt’s account tally with the fact that for seven years Newton was effectively left in Woolsthorpe to be brought up by his maternal grandmother.” — From ‘Newton: A Very Short Introduction’ by Robert Iliffe
Peter Simonini practices at the Sullivan School of Irish Dance in Newton. Ten-year-old Simonini is headed to Scotland this March to compete in the World Championships of Irish Dance.
Once more, Ray disappears for the rest of the chapter in the comic so here, have some domestic doodles of Ray and his emotional support boyfriend, Newt (whomst belongs to @shylittlemoosen uwu <33 ), all from the last few weeks
Muchos creen que tener talento es cuestión de suerte, nadie piensa que tener suerte es cuestión de talento.
Leonardo da Vinci.
Mientras le preparaban la cicuta, Sócrates leía una melodía para flauta (una aria)
— ¿De qué te va a servir? — Le preguntaron.
— “Para saberla antes de morir".
Corría el año 1922, Einstein acaba de ganar el premio Nobel de Física. Un niño le dice a su madre que quiere ser investigador, pero le preocupa que, al paso que va la ciencia, cuando sea mayor ya no quede nada por descubrir. Años después el niño se ha licenciado en física, pero interrumpe su doctorado cuando una bomba nazi destruye su laboratorio. Se incorpora entonces al servicio secreto británico y diseña una mina especial para hundir los dragaminas alemanes. Inconforme aún con su vida, incursiona en otro campo, y decide descubrir el secreto de la vida. Con un hatajo de visionarios inaugura la era del genoma y gana el Premio Nobel de Medicina por este trabajo. A los 60 años decide que el último territorio que queda por explorar para comprender la vida es la consciencia. A la edad en que la mayoría de la gente está pensando en la jubilación, él empieza una nueva carrera como neurocientífico. Durante casi treinta años genera ideas y ejerce una poderosa influencia, como pocos otros científicos de su tiempo. Pocas horas antes de morir, en el 2004, Francis Crick termina de corregir un manuscrito para los investigadores futuros que quieran entender mejor qué es la consciencia.
La suerte sólo favorece a la mente preparada.
Louis Pasteur.
Si incluso los cerebros más privilegiados y las personas con una capacidad de trabajo extraordinaria se mueren sin saberlo todo, ¿qué esperanza nos queda a los que tenemos capacidades más ordinarias?
No nos queda otro remedio que admitirlo: no podemos saberlo todo. Lo máximo a lo que podemos aspirar es a saber algunas cosas, pero a saberlas bien.
Hoy sabemos mucho más de astronomía que Ptolomeo o Kepler, de física que Newton e incluso Einstein, de medicina que Hipócrates, de química que Lavoisier. Si vemos más lejos es porque estamos subidos en hombros de gigantes. Nuestra medida del universo es más exacta que la de Copérnico. A pesar de lo que no sabemos y de lo que no nos imaginamos que no sabemos, podemos decir que el cúmulo de conocimiento que tenemos es mayor, objetivamente mayor, que el que se tenía en la antigua Grecia, o incluso hace dos siglos. Es la historia del esfuerzo intelectual del hombre por comprender el mundo en el que le tocó vivir.
Hace apenas unos cuantos siglos, no teníamos la menor idea del lugar que ocupábamos en el universo, dónde estábamos, cuándo estábamos, nos encontrábamos perdidos en una especie de prisión.
Éramos cazadores y recolectores, la frontera estaba por todos lados, sólo nos limitaba la tierra, el océano y el cielo. Pero rompimos las cadenas de esa prisión. Fue el trabajo de generaciones de incansables buscadores, ellos cuestionaron la autoridad, empezaron a pensar por si mismos, a cuestionarse así mismos. Trabajaron y probaron sus ideas por medio de la evidencia obtenida a través de la observación y la experimentación y nunca se olvidaron de recordar que podrían estar equivocados.
Toda nuestra ciencia comparada con la realidad es primitiva e infantil, y sin embargo es lo más preciado que tenemos.
Voltaire dijo de los hombres de su tiempo que su grandeza consistió en que necesitaban milagros y simplemente los hicieron.
El hombre ha llegado a atisbar en la enormidad del universo y en su insólita complejidad, y ha tenido que admitir con valentía y cierta decepción, que su lugar en el escenario total es insignificante. Pero aún así no se amedrentó y continuó su búsqueda.
Nuestra aventura actual es más asombrosa que cualquier novela, ahora podemos ver a voluntad cosas que antes sólo eran posibles en los sueños.
Si miramos al pasado, muchos de los grandes inventores, no fueron los primeros en concebir la idea, pero si fueron los primeros en hacerla posible, ellos son los que figuran en los libros de historia. Imaginar es de sabios, hacer es de genios. En perspectiva, todo depende del precio que estemos dispuestos a pagar.
Imaginen toda la vastedad del universo, una inmensidad de espacio y tiempo, una vastedad mayormente inexplorada. Imaginen cuántos secretos esconde, cuántos misterios aguardan por nosotros. La ciencia nos puede llevar por toda esa grandeza y nos puede revelar esos misteriosa, pero sin imaginación no vamos a ningún lado. Todo cuanto podamos llegar a imaginar está impulsado por dos motores: escepticismo y asombro, y se guía por el conjunto de normas sencillas que rigen la ciencia y la hacen tan poderosa. Probar ideas con experimentos y observación, edificar en esas ideas que pasen la prueba y desechar las que no la pasen. Seguir la evidencia hasta donde nos lleve y cuestionarlo todo. El hombre ha tomado esas reglas en serio y ha puesto el cosmos a sus pies.
El hombre sabe al fin que está solo en la inmensidad indiferente del universo, de donde ha surgido por azar. Su deber, como su destino, no está escrito en ninguna parte, le corresponde a él elegir entre el reino trascendente de las ideas y del conocimiento, o el de las tinieblas.
¿Qué tan lejos habrá deambulando nuestra especie de nómadas a finales del próximo siglo y a finales del próximo milenio?
Con todos nuestros defectos, a pesar de nuestras limitaciones y falibilidades, nosotros los humanos somos capaces de la grandeza.
Muchos se maravillan ante la enormidad de una montaña, ante el poder de los mares tempestuosos, o ante la grandeza del firmamento en una noche clara. Pero pasan de largo sin maravillarse, sin sorprenderse de sí mismos y de sus compañeros de especie.
Estamos hechos del mismo material del que están hechas las estrellas, pero hay hombres tan grandes como esas estrellas de dimensiones ciclópeas, destinados a arder para que la tierra pueda ser iluminada.
El camino abierto sigue llamándonos suavemente como una canción casi olvidada de la infancia.
La ignorancia no es decir: no lo sé, ignorancia es no querer saberlo.
Mi admiración y agradecimiento para @buckhead1111 por su trabajo hermoso e impecable y por compartirlo con todos nosotros.
La primera imagen de esta publicación hace parte de su exquisito trabajo.
dont talk to me if you don’t know that this, recruits, is a 20-kilo ferrous slug. Feel the weight! Every five seconds, the main gun of an Everest-class dreadnought accelerates one to 1.3 percent of light speed! It impacts with the force of a 38-kiloton bomb! That is three times the yield of the city buster dropped on Hiroshima back on Earth. That means: Sir Isaac Newton is the deadliest son of a bitch in space. Now! Serviceman Burnside! What is Newton’s First Law?
An object in motion stays in motion Sir!
No credit for partial answers, maggot!
Sir! Unless acted upon by an outside force, sir!
Damn straight! I dare to assume you ignorant jackasses know that space is empty! Once you fire this hunk of metal, it keeps going till it hitssomething. That can be a ship. Or the planet behindthat ship. It might go off into deep space and hit somebody else in ten thousand years! If you pull the trigger on this, you are ruining someone’sday,somewhereandsometime.Thatis why you check your damn targets! Thatis why you wait for the computer to give you a damn firing solution! Thatis why, Serviceman Chung, we do not “eyeball it!” This is a weapon of mass destruction! You are nota cowboy shooting from the hip!
![Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]](https://64.media.tumblr.com/0a261cdbbe280d83c3e5cef5927ffaf6/tumblr_p8xj0g5MH51rfw53wo2_1280.jpg "Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]")
![Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]](https://64.media.tumblr.com/615aedbfef84fd9ae423d8b38ba2d5a5/tumblr_p8xj0g5MH51rfw53wo3_1280.jpg "Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]")
![Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]](https://64.media.tumblr.com/761e76dc3c2628f85cdec2485c6640b1/tumblr_p8xj0g5MH51rfw53wo4_1280.jpg "Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]")
![Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]](https://64.media.tumblr.com/2238a9b9fc68a628250a90b76b8fbdb5/tumblr_p8xj0g5MH51rfw53wo6_1280.jpg "Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]")
![Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]](https://64.media.tumblr.com/c03c579e1f9a3094d06d30dda82d1ba9/tumblr_p8xj0g5MH51rfw53wo5_1280.jpg "Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]")
![Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]](https://64.media.tumblr.com/ecae689324e534f62ce90cd03f529f71/tumblr_p8xj0g5MH51rfw53wo1_1280.jpg "Newton’s annual Kids Fun Fest at City Hall on May 13, 2018. [Wicked Local Photo/Ruby Wallau]")
![Mind Map #47: [Tragedy reminds us to love] *Note: The drawing of the man and child is loosely based](https://64.media.tumblr.com/4dd6ca2f79e820abf2c145255d03f106/tumblr_mf3y60p8Ui1ru4tgjo1_1280.jpg "Mind Map #47: [Tragedy reminds us to love] *Note: The drawing of the man and child is loosely based")
