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from the Arabic 'al-jabr' literally meaning the "reunion of broken parts". If not Algebra, some kind of math problem....
_IMG3872
Oh, algebra! I could liked you, if you was more simple :))
Description and conditions Project 365 - www.flickr.com/photos/danchee/sets/72157628234157885/
Definitions as found in Webster’s New World Dictionary of the American Language—College Edition. Cleveland, Ohio and New York, New York: The World Publishing Company. ©1966.
This dictionary was given to me for my birthday by my parents in 1967 when I was in 8th grade. This will be the source of my information for this year's FAFM project.
1. A Algebra (al’-jƏ-brƏ) noun. [Italian < Arabic al-jebr, the reunion of broken parts < al the + jabara, to reunite] 1. The branch of mathematics that uses positive and negative numbers, letters, and other systematized symbols to express and analyze the relationships between concepts of quantity in terms of formulas, equations, etc.; generalized arithmetic; abbreviated alg. 2. A textbook or treatise dealing with the branch of mathematics..
I was a math major in college, and algebra was one of my favorite subjects. Years later, I helped tutor a number of students after school, and this index card was used to help them remember the "Quadratic Equation". The four steps were an inspiration to help them remember the process on solving these equations. Of course, they also had to know the slope-intercept formula, y = mx +b.
Okay, too much information at one time, but don't worry. my 2024 FAFM will not be all math! More fun topics to come! (And yes, some of my Lego Minifigures will also be assisting me!)
When I took the picture I saw a potential abstract from reality... When I opened it in Photoshop I found a Mensa-style puzzle:
3 (boards), 3 (Tacks), 2 (lines), 12 (staples)... what is the next number in the sequence?
I have no idea, I just take pictures...
I then noticed the algebraic equasion:
(3+3)2 = 12
Now I will toss and turn all night trying to work in the 4 corners...
*sigh* I should have simply deleted the silly thing.
7 days of shooting
week #2
birds
geometric Sunday
In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics.
The two precise formulations of the theorems differ somewhat. The original theorem, as stated by Jean-Pierre Serre in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closed field (of any characteristic). The complementary variant stated by Richard Swan in 1962 is more analytic, and concerns (real, complex, or quaternionic) vector bundles on a smooth manifold or Hausdorff space.
Born in Khiva around 780 CE, al-Khorezmi (or al-Khwarizmi meaning 'of Khwarizm', the original name of Khiva) was a great mathematician, scientist and author. He is considered the father of algebra (the word derives from the name of his book on the subject, Al-Jabr wa-al-Muqabilah, which became the standard mathematical book at European universities until the 16th century. He made an even more important contribution (hard to imagine….) in developing the concept of an algorithm. In this capacity, some consider him to be the grandfather of computer science. The word algorithm is derived from a Latin corruption of his name.
In addition to his original work, al-Khorezmi also synthesized the work of other great scholars of the era. After reviewing various numerical systems, he adopted the Hindu system (in the western world, we incorrectly call it the Arabic system) of numerals and was the first to recognize the importance of the number zero. It was through al-Khorezmi's work that this numerical system spread to the Middle East and then Europe and the rest of the world.
Beyond mathematics, al-Khorezmi made contributions to the field of geography by supervising the creation of a map of the world and to astronomy where he wrote about clocks, astrolabes and sundials.
At the time, Khiva was part of Persian Khorasan that was under the auspices of the Arab Abbasid caliphate administered from Baghdad. Al-Khorezmi moved to Baghdad early in his life and studied there at the House of Wisdom, a scientific research and teaching center.
Khiva, Uzbekistan
This photo is dedicated to good friend and scholar, O Bejeweled Land (Zendeh baad Azadi).
Explore #186 on July 26, 2010
It is impossible for me to understand how I have so much trouble helping him with algebra. I used to ace math. 30 years later I am struggling when I try to help him.
As phoebird requested, here are details of the triptich. A fine idea for flickr because one of the problems here is getting a sense for larger pieces that I do. This triptich will be about 18" x 48" on the wall, if not a little lerger. But I kind of like them as iindividiual statements too.
Known for Abstract algebra, Amalie Emmy Noether; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's theorem, which is fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed some theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent.
Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called the "Noether boys". In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas; her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra. By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world. The following year, Germany's Nazi government dismissed Jews from university positions, and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania where she taught, among others, doctoral and post-graduate women including Marie Johanna Weiss, Ruth Stauffer, Grace Shover Quinn and Olga Taussky-Toddone. At the same time, she lectured and performed research at the Institute for Advanced Study in Princeton, New Jersey.
Noether's mathematical work has been divided into three "epochs". In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields. Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra". In her classic 1921 paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains), Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications. She made elegant use of the ascending chain condition, and objects satisfying it are named Noetherian in her honor. In the third epoch (1927–1935), she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology
Born: Amalie Emmy Noether 23 March 1882 Erlangen, Bavaria, German Empire
Died: 14 April 1935 (aged 53)
Bryn Mawr, Pennsylvania, United States
en.wikipedia.org/wiki/Emmy_Noether
Orginal Photo: en.wikipedia.org/wiki/Emmy_Noether#/media/File:Noether.jpg
Artwork by TudioJepegii
I Grok Matrix Theory and Linear Algebra. I found this copy of my fatherinlaw's "world renowned" book on Linear Algebra he wrote in the 1960s on eBay. At $8 and an old library copy we got it for the cheap entertainment of notes, where it had been (England) and other smalls.
Turn your eyes inside and dig the vacuum.
www.floridamemory.com/items/show/261088
6th grader Lenora Humphry and Sylvia Stabler doing algebra at Kate Sullivan School in Tallahassee, Florida.
Image Number
TD00424
Date
1957
Date Note
Photographed on November 18, 1957.
Geographic Term
Leon County
Tallahassee
Subject Term
Public institutions--Florida--Leon County--Tallahassee
Public schools--Florida--Leon County--Tallahassee
Elementary schools--Florida--Leon County--Tallahassee
Mathematics--Study and teaching (Elementary)
Algebra--Study and teaching (Elementary)
School children--Florida--Leon County--Tallahassee
Girls--Florida--Leon County--Tallahassee
Personal Subjects
Humphry, Lenora.
Stabler, Sylvia.
"Algebra applies to the clouds, the radiance of the star benefits the rose--no thinker would dare to say that the perfume of the hawthorn is useless to the constellations. Who could ever calculate the path of a molecule? How do we know that the creations of worlds are not determined by falling grains of sand? Who can understand the reciprocal ebb and flow of the infinitely great and the infinitely small, the echoing of causes in the abyss of being and the avalanches of creation?"
— Victor Hugo (Les Miserables)
Life's ebb and flow... good times and bad times come and go!
Strobist: AB800 with Softlighter II camera right. AB800 open behind backdrop of white faux suede. Triggered by Cybersync.
Joke outside a St. Pete florist, and a reminder of a school subject that I utterly loathed (and had no use for in my adult life).
Finn and I got to meet Jeremy Shada and Pen Ward!! They signed the back of Finn's head and she is LOVING her new ink!! Too damn cool. ❤
" We2=1 (2+1=3) The Triple concept " by White Angel.
Lettering with silver inks on black paper. June 22 2015.
Ref. 3054
From the author: "Conceptually, if you and me are two, the product of our respective components is Us, so, alltogether we are 3 Algebrically, the formula seems to match.
My first alphanumeric lettering has been made for & dedicated to my Micio because We2are1 ;-) ."
White Angel aka łwAE
Kind attn. Antibotteurs Group: I mean this post now for this specific group as an invite to let us post 3 pictures a day. In fact artists like to post subsequent pics belonging to the same collection. Also, conceptually, 2 or 3 pics can be necessary for topics/ideas/hints on confrontation different photos or artworks while fronting a certain concept. Thanks for your consideration!
#Aritmancy #Algebra #AlgebricFormulas
When I was just a young lad I had what many of us fear the most... The Evil Math Teacher! She was a wicked, mean, nasty old ancient creature who seemed to take an unusual amount of pleasure in our not having any. I'm not sure why she was like that, perhaps the man she most loved ran off with another (understandably, probably became an Astronaut, minimum safe distance...), perhaps she had no friends or a poor upbringing or maybe, she was just mean and always had been. Some people are just like that, hiding some inner defiency under a coat of ugly I guess. I didn't wonder about that back then, I wondered how I was ever going to do well in this class, math is a difficult subject enough. I mean what the devil is the point of treating a bunch of young minds so poorly? About a month into the school year Lady Luck heard our prayers and smiled upon us, The EMT (Evil Math Teacher) had gotten herself pinched between two cars and had to have knee surgery to repair the damage. She would miss a majority of the year! I'm not one to celebrate others misfortunes- usually.
Our substitute teacher was a kind middle aged fella who was semi-retired as I recall. Good natured, soft spoken and even tempered, he was most welcomed by the class and as it turned out he was a pretty darned good teacher to boot. I was running a 'C' average with the female Lord Voldemort at the helm but after a few weeks with the substitute, my weekly quiz scores climbed steadily. The year passed by quickly but good fortune would not last, rumours began of her return.
The week of the finals rumour became reality and the wicked woman was there and passed out our tests, her knee apparently repaired, her character still broken, not too quietly she grumbled as we worked through to the end. We went home for the weekend and waited. Monday morning EMT returned our tests with the scores written in red ink at the top. My results, to her dismay and maybe a little of mine - was a 93! Far too good of a score for her to believe unfortunately, if I had passed with a 70 she wouldn't have thought twice, but a 93? No way was that going to fly. I was accused of cheating and nothing I would say could make her think otherwise, if I hadn't cheated then was not her teaching ability in question? I would have to retake the test... Alone... Just her... And I... After school... Two months of meanness bottled into one foul release. Fine! If it's a gunfight you want lady, you've got it. For once in my small, simple little life I'm ready, I'm prepared.
The end of the day found us together, our desks facing each other, her spectacled scrunchy eyes peering at me intently as I sat and retook the test, every movement observed. What? No pat down? The questions were the same but in different order, enough to throw me off if I had them written down inside my arm. The big white clock punched out the seconds in its usual distracting manner. On the thin side of 45 minutes later 'It' informed me time was up. She swept in and snatched the papers from my desk, and graded it on the spot. What could almost be called a good mood she finally displayed, surely thinking she had me now! But to my great satisfaction whatever inkling of happiness she was on the verge of was taken from her. My lips pursed at the corner into my best Clint Eastwood sneer ever. Her body convulsed as my bullets flew into her, hers missed me completely. The dust settled. It was over. The music could swell up now, for here at the end of the year, my second final test, I scored a 97!
lgebra Tea House is that quirky little Bohemian cafe on Murray Hill Road, with the bright umbrellas and free-form garden of flowers and herbs.
Cyan, magenta, and yellow filters, arranged on light table like a Venn diagram, make red, blue, and green colors where they overlap. Magenta and yellow transmit red light. Magenta and cyan allow blue to pass. Cyan and yellow passes green. All filters together block all light. These are examples of the subtractive nature of colored filters. R-B-G, the usual color space of digital media, is thusly obtained from C-M-Y, which is color space of printed media.
When red, blue, and green filters are arranged similarly, observe that each colored filter does not transmit any other color except itself.
Boolean algebra gleaned from Venn diagrams symbolically:
for R-B-G
(R ^ B) v (B ^ G) v (G ^ R) = K, where K is black
for C-M-Y
M ^ Y ^ -C = R, M ^ C ^ -Y = B, C ^ Y ^ -M = G, C ^ M ^ Y = K
9th Grade Classroom, Peery Matriculation School, Thottanoval, Tamil Nadu
Students effectively using technology in their 9th grade Algebra class. This class mixed formal math instruction with using computers in the classroom. We were teaching the children to use computers in the classroom and in particular helping them use Khan Academy (www.khanacademy.org/) in this classroom. Here, students are watching video lessons after the teacher went through the lessons of the day.
This school is run by Rising Star Outreach (RSO), a non-profit organization working to eliminate leprosy in India. You can read more about this organization here.
Children studying in this school are primarily from nearby leprosy colonies and villages around the school. If you are looking for a good cause and are interested in supporting children's education, please consider RSO. This is real and they are doing some truly amazing work.
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Brothers Brick posted a review on the upcoming 70922 The Joker Manor set (comes out on Black Friday). I kinda scrolled through the page and looked at the pictures assessing whether or not I want to spend the time reading. I passed because there’s just… too much. And the thing is,...
www.fbtb.net/lego/2017/11/22/sometimes-reviews-can-be-too...
Kept back by the deeper rumblin's in my breast
Bad vibes puttin' mad pressure on my chest
Fly time is a rhythm clockin' exocet
Mad lines from a stranger you've already met
~Soul Hooligan~
How Not to Square the Circle
Nicholas of Cusa was attacking a problem dating back to the ancient Greeks. The solution would have made him famous forever...
Tony Phillips
Stony Brook University
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Introduction
In 1965 my late friend and colleague John Stallings wrote "How not to prove the Poincaré conjecture." This work appeared in the Proceedings of the Wisconsin Topology Seminar and is still available on John's Berkeley website. It begins with the declaration: "I have committed the sin of falsely proving Poincaré's Conjecture. But that was in another country; and besides, until now no one has known about it." It continues with the exposition of the main ideas relating the conjecture to statements in algebra, and is certainly what Stephen Miller had in mind when he wrote, in the AMS Notices, after John's death, "His 1960s papers on the 3-dimensional Poincaré Conjecture are both brilliant and hilarious at the same time."
In 1445 Nicholas of Cusa wrote De Geometricis Transmutationibus (On Geometric Transmutations); my account is based on a recent translation into French of all of Nicholas' mathematical works: Nicolas de Cues, Les Écrits mathématiques by Jean-Marie Nicolle (Honoré Champion, Paris, 2007). This was the first of Cusa's writings in which he addressed the problem of squaring the circle. Literally, squaring the circle means devising the straightedge-and-compass construction of a square whose area equals that of a given circle. This means a construction relating a segment of length 1 (the radius of the circle) to a segment of length π√ (the side of the square). Nicholas's plan was start from an equilateral triangle and construct an isoperimetric circle; this is the content of the First Premise in De Geometricis Transmutationibus. If the triangle had perimeter 1, the circle would have diameter 1/π. Then the composition of two more standard straightedge and compass constructions could start from that diameter and generate first a segment of length 1/π−−−√, and from that one a segment of the reciprocal length π√.
Examples of straight-edge and compass arithmetic:
Left: square root. A segment AB of length x is extended by a segment BC of length 1. Choose one of the points D where the (green) circle with diameter AC intersects the perpendicular through B. Then by plane geometry DB2=AB⋅BC=AB, so DB has length x√.
Right: reciprocal. The construction starts with a segment EF of length 2 extended by FG of length 12. A circle (green) is constructed with EG as diameter. For any x between 1/2 and 2, for example 1/π−−−√, a circle (blue) of radius x is drawn with center F. Choose one of the intersection points X of the two circles and draw the line through X and F. It will intersect the green circle at a second point Y; the length y of the segment FY will be the reciprocal of x, since by standard plane geometry XF⋅FY=EF⋅FG=1.
Other circles, lines and points used in the constructions are shown in black.
Nicholas of Cusa
Nicholas of Cusa (1401-1464) was one of the leading intellectual figures in early 15th-century Europe. He is often described as a transitional figure between the Middle Ages and the Renaissance, and in fact he was personally involved in one of the great events that mark that transition: Pope Eugene IV sent him to Constantinople in 1437 as part of a delegation to negotiate the participation of the Eastern Orthodox hierarchy in the Council of Florence. They came, with an entourage of distinguished Greek scholars who stayed, and lectured, in Florence; contributing to the surge of interest in humanistic learning which led to the new age.
Nicholas' principal occupations were ecclesiastical politics and administration (he was named Cardinal in 1449) and, relatedly, theology/philosophy. Those were tumultuous times for the Church; Nicholas was at the center both of bitter jurisdictional controversies and of intense disputation about the exact wording of dogmatic texts, where the placement of a comma could assume cosmic importance. In those days philosophy, theology and natural science were closely linked: the physical structure of the universe had deep theological implications. Nicholas' energetic and erudite mind, in a priori meditation, led him to scientific insights that turned out to be prophetic. For example, he understood that the earth, the sun and the moon were objects moving through space; and he rejected the idea that all orbits had to be circular or even that the universe had a center (De Docta Ignorantia, Book II). Here he was a predecessor of Kepler (who referred to him as "divine") and of Giordano Bruno.
Nicholas' interest in mathematics seems to have been its status as an impregnable logical system. He believed that by testing his philosophical theories in mathematics he could produce convincing evidence of their validity. He outlines the parallelism between geometry and theology in De Circuli Quadratura, dated July 1450. "Transport yourself by assimilation from these mathematics to theology. ... Just as the circle is perfection in a figure, since any perfection of figures is worked into it, its surface contains all the surface of all figures and has nothing in common with all the other figures, but is absolutely one and simple in itself; likewise absolute eternity is the form of all forms ... having nothing in common with any other form. And whatever the figure of the circle therein may be, since it has neither beginning nor end, it has resemblance with eternity ... . ... Likewise, if a triangle wanted to triangulate the circle, or a square to square it and so forth for the other polygons, thus also intellectual nature wants to understand [God]."
The First Premise and its "proof"
Nicholas of Cusa's First Premise: a is the center of the equilateral triangle bcd. "You divide the side bc into four equal parts which you mark e,f,g: I assert that, if one extends the line drawn from a to e by its fourth, which gives ah, this will be the radius of the circle whose circumference is equal to the three sides of the triangle."
One of the thought schemata Nicholas devised for use in theology was the "concidence of opposites." Here is how he applied the principle to the proof of his First Premise. The construction involves a parameter, namely the position of the point e on the line cb. Nicholas observes that when e is at the midpoint f the length of the segment ah is smaller than the desired radius, and that when e is at b the length is larger. He applies the principle: ubi est dare magis et minis, quod ibi sit dare aequale (where one can give a greater and a lesser, one can also give an equal; essentially the Intermediate Value Theorem) and concludes correctly that for some intermediate position x the length ah must be exactly equal to that radius, "and that is the point e equidistant between b and f." The last statement made with no justification.
The construction is in fact plausible: suppose the sides of the triangle have length 1. Then ef=14; similarity of triangle abf with a half-equilateral triangle, and the Pythagorean theorem, yield af=123√; so ae=748−−√, and ah=(5/4)748−−√; the First Premise states that 2π⋅ah=3, which implies π=65487−−√=3.1423... . This value, which Nicholas could have calculated but never mentions explicitly, was within the bounds [22371,227]=[3.14084...,3.14285...] established by Archimedes. Therefore, until better approximations to π were available, there was no way to prove Nicholas's construction wrong, even though there were obvious gaps in his proof.
Later developments: Things get worse
Nicholas circulated copies of his work among his friends, who included Paolo Toscanelli (1397-1482), a Florentine astronomer and physician. He had been Nicholas' classmate, and they remained good friends for life. Toscanelli wrote back with objections. To us, now, it is clear that there was no way the argument could be repaired. Nicholas' solution was to devise a different, and considerably more complicated, construction.
The diagram for Nicholas of Cusa's second quadrature construction, from Quadratura Circuli, 1450. The construction starts from a triangle cde, superimposes an isoperimetric square ilkm and yields rq as the radius of the isoperimetric circle.
Nicholas would have done better to stay with his first construction. The new one was reprinted and minutely analyzed by Regiomontanus (Johannes Müller, 1436-1476) who showed that the implied value for π was outside the Archimedean bounds (Nicolle calculates it as 3.154); this is part of a 60-odd page appendix to his De triangulis omnimodis, dated 1464, published in 1533. There Regiomontanus takes up all of Nicholas' constructions one by one and "does the math" (Nunc ad numeros descendendum), using his knowledge of trigonometry to show "that Nicholas' approximations to π were --except one-- not even within the limits established by Archimedes," according to Menso Folkerts, who characterizes Regiomontanus as "a gifted student of Archimedes," and Nicholas of Cusa as "an amateur in mathematics." The one exception is presumably the First Premise above.
The moral of the story
Nicholas of Cusa was attacking a problem dating back to the ancient Greeks. The solution would have made him famous forever, and might even have helped bolster his side in theological disputations. No one knew at the time that squaring the circle is impossible: the proof requires calculus, which was 200 years away; and even then it was not discovered until 1882.
John Stallings was also attacking a famous problem: 50 years old, a very long time in modern mathematics. In this case the problem was not impossible, but the methods that led to its solution lay far in the future. Richard Hamilton's introduction of the Ricci flow, which led to Gregory Perelman's ultimate victory, came out in 1982, some 17 years after Stallings wrote "How not to prove the Poincaré Conjecture." But Stallings discovered his error by himself, before publishing, whereas Nicholas seems to have believed until the end that he had squared the circle, but perhaps had not been able to find the right argument to substantiate his claim.
Here is how Stallings ends his story: "... I was unable to find flaws in my 'proof' for quite a while, even though the error is very obvious. It was a psychological problem, a blindness, an excitement, an inhibition of reasoning by an underlying fear of being wrong. Techniques leading to the abandonment of such inhibitions should be cultivated by every honest mathematician."
Why circle-squaring is impossible
We will see that any length occurring in a compass and straightedge construction starting from length one must be an algebraic number, i.e. it must be a root of a polynomial with integer coefficients. Considerably more intricate is the proof that π, and therefore π√, is transcendental, i.e. not algebraic. Some references are given here.
A random compass-straightedge construction: all the coordinates of the vertices produced by the construction are of a special form: they are obtained from 1 by composing a finite number of operations, which can be arithmetic (sum, product, quotient, etc.) or the extraction of a square root. For future use, let's call the set of these numbers S. In this example, the construction starts with the vertices O=(0,0) and A=(1,0); the line they span is the x-axis. The circle of center O and of radius OA intersects the circle of center A and of radius OA at B=(12,3√2), the x-axis at C=(−1,0) and the y-axis (the perpendicular bisector of AC, constructed as usual by two circles and a line) at E=(0,1). The circle of center E and radius EA intersects the line through O and B at D=(3√ 7√4,3 21√4). The circle of center A and radius AD intersects the x-axis at F=(1 AD,0)=(1 127 21−−√−3√−7√−−−−−−−−−−−−−−−−√,0). As the construction continues, the number of embeddings of radicals into radicals tends to rise, but the numbers always have this general form. They are clearly algebraic, since the radicals can be peeled off by continued squaring and rearranging. In fact these constructible numbers form a special class of algebraic numbers: those that can be reached from the rational numbers by a finite number of quadratic extensions, i.e. by arithmetic operations and taking square roots a finite number of times. To show squaring the circle is impossible, "algebraic" is sufficient; but other impossibilities (duplicating the cube, trisecting the angle) require this additional information.
To see why this works in general, note first that if points P and P′ have their coordinates in S, then by the Pythagorean theorem their distance PP′ = r must also belong to S. So the circle of radius PP′ about P, say, has the equation (x−p1)2 (y−p2)2=r2. Another circle constructed from two points with coordinates in S will have a similar equation, say (x−q1)2 (y−q2)2=s2. All these coefficients lie in S. The coordinates of the intersection points of the two circles (if they intersect) will be the pairs (x,y) satisfying both equations. From the first equation we can write y=±r2−(x−p1)2−−−−−−−−−−−√ p2. Substituting this value in the second equation yields a polynomial equation in x; it looks like it might have degree 4, but the higher powers cancel and it is a quadratic equation with coefficients in S. The quadratic formula involves arithmetic and a square root, so the solutions it produces will again belong to S. For intersections of a circle and a line no cancellation is needed; the equation is quadratic; and for the intersection of two lines it is linear.
Squaring the circle
From Wikipedia, the free encyclopedia
For other uses, see Square the Circle.
Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.
Impossibility[edit]
The solution of the problem of squaring the circle by compass and straightedge demands construction of the number , and the impossibility of this undertaking follows from the fact that pi is a transcendental (non-algebraic and therefore non-constructible) number. If the problem of the quadrature of the circle is solved using only compass and straightedge, then an algebraic value of pi would be found, which is impossible. Johann Heinrich Lambert conjectured that pi was transcendental in 1768 in the same paper in which he proved its irrationality, even before the existence of transcendental numbers was proven. It was not until 1882 that Ferdinand von Lindemann proved its transcendence.
The transcendence of pi implies the impossibility of exactly "circling" the square, as well as of squaring the circle.
It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of pi, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.
Bending the rules by allowing an infinite number of compass-and-straightedge operations or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can be in Gauss–Bolyai–Lobachevsky space. Indeed, even the preceding phrase is overoptimistic.[7][8] There are no squares as such in the hyperbolic plane, although there are regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent (but these angles are strictly smaller than right angles). There exist, in the hyperbolic plane, (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area. However, there is no method for starting with a regular quadrilateral and constructing the circle of equal area, and there is no method for starting with a circle and constructing a regular quadrilateral of equal area (even when the circle has small enough radius such that a regular quadrilateral of equal area exists).
Some apparent partial solutions gave false hope for a long time. In this figure, the shaded figure is the Lune of Hippocrates. Its area is equal to the area of the triangle ABC (found by Hippocrates of Chios).
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Squaring the circle Basel problem Feynman point Other topics related to π
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Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.
The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.[1]
The term quadrature of the circle is sometimes used synonymously or may refer to approximate or numerical methods for finding the area of a circle.
Liu Hui
Mathematical work[edit]
Along with Zu Chongzhi (429–500), Liu Hui was known as one of the greatest mathematicians of ancient China.[1] Liu Hui expressed all of his mathematical results in the form of decimal fractions (using metrological units), yet the later Yang Hui (c. 1238-1298 AD) expressed his mathematical results in full decimal expressions.[2][3]
Liu provided commentary on a mathematical proof identical to the Pythagorean theorem.[4] Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".[5]
In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry. For example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge.[6] He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. In his commentaries on the Nine Chapters, he presented:
An algorithm for calculation of pi (π) in the comments to chapter 1.[7] He calculated pi to with a 192 (= 64 × 3) sided polygon. Archimedes used a circumscribed 96-polygon to obtain the inequality , and then used an inscribed 96-gon to obtain the inequality . Liu Hui used only one inscribed 96-gon to obtain his π inequalily, and his results were a bit more accurate than Archimedes'.[8] But he commented that 3.142074 was too large, and picked the first three digits of π = 3.141024 ~3.14 and put it in fraction form . He later invented a quick method and obtained , which he checked with a 3072-gon(3072 = 512 × 6). Nine Chapters had used the value 3 for π, but Zhang Heng (78-139 AD) had previously estimated pi to the square root of 10.
Gaussian elimination.
Cavalieri's principle to find the volume of a cylinder,[9] although this work was only finished by Zu Gengzhi. Liu's commentaries often include explanations why some methods work and why others do not. Although his commentary was a great contribution, some answers had slight errors which was later corrected by the Tang mathematician and Taoist believer Li Chunfeng.
Survey of sea island
Liu Hui also presented, in a separate appendix of 263 AD called Haidao suanjing or The Sea Island Mathematical Manual, several problems related to surveying. This book contained many practical problems of geometry, including the measurement of the heights of Chinese pagoda towers.[10] This smaller work outlined instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed at right angles to them".[11] With this, the following cases are considered in his work:
The measurement of the height of an island opposed to its sea level and viewed from the sea
The height of a tree on a hill
The size of a city wall viewed at a long distance
The depth of a ravine (using hence-forward cross-bars)
The height of a tower on a plain seen from a hill
The breadth of a river-mouth seen from a distance on land
The depth of a transparent pool
The width of a river as seen from a hill
The size of a city seen from a mountain.
Liu Hui's information about surveying was known to his contemporaries as well. The cartographer and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a rectangular grid and graduated scale for accurate measurement of distances on representative terrain maps.[12] Liu Hui provided commentary on the Nine Chapter's problems involving building canal and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.[13]
Although translated into English long beforehand, Liu's work was translated into French by Guo Shuchun, a professor from the Chinese Academy of Sciences, who began in 1985 and took twenty years to complete his translation.
Zu Chongzhi
The majority of Zu's great mathematical works are recorded in his lost text the Zhui Shu. Most scholars argue about his complexity since traditionally the Chinese had developed mathematics as algebraic and equational. Logically, scholars assume that the Zhui Shu yields methods of cubic equations. His works on the accurate value of pi describe the lengthy calculations involved. Zu used the Liu Hui's π algorithm described earlier by Liu Hui to inscribe a 12,288-gon. Zu's value of pi is precise to six decimal places and for a thousand years thereafter no subsequent mathematician computed a value this precise. Zu also worked on deducing the formula for the volume of a sphere.
so this shoot was interesting. this was taken on my roof from the windowsill. i tried this shoot twice: once when it was raining, and once when it was not. the second one worked out waaayyy better, even though i didn't get too many shots because i forgot my memory card was still full from the other night. this looked sooo bad in the original. but i liked the idea, so i edited it like the bad person that i am. oops! i'm breaking my sooc streak.
anyway. yeah. today was pretty good. my birthday is in 47 days. me and maddie are gonna plan an awesome party. it's been approved by the mother. i'm pumped. like beyond pumped. this is gonna be so exciting it's riciddd. my mom really likes the idea too. i mean whaattt! what parent actually likes party ideas. mine apparently! i'm so freaking exciteddd!
today was pretty good. i can't play soccer still. it was really awkward cause my old school played my new school in field hockey and i was watching and i was cheering for my new school and yeah... it was just awkward. whatever. school was fun though. especially the party planning party. that was rockinnn. we had a "surprise" quiz in algebra that we hadn't been told about but it was on our syllabus and we were like whaattttt. i don't even know. watching soccer/field hockey was fun though. then i came home and ate a lot of yummy food. i swear we have the tastiest grapes at my house right now. seriously. SO GOOD.
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"La Muerte es como una ecuación algebráica, una serie de factores "x" y valores "y", sumados y multiplicados, divididos y restados que al final dan una solución simple y definitiva = 0"
"Death is like an algebraic equation, a number of "x" and "y" factors, added and multiplied, divided and subtracted that in the end result into a simple and final = Ø"
More ceiling detail from the Getty Museum with false color added by Picnik. The purple seemed to make the most visually striking overlay.
You may recognize what this is, but it's presented in a "non-representational" way, in that the colors and the shapes and the underlying pattern are the focus of the picture and not the symbols themselves. I believe this would be classified as Abstract.
1. Abstract. From wikipedia: Abstract art is now generally understood to mean art that does not depict objects in the natural world, but instead uses color and form in a non-representational way. You are free to create your abstraction entirely in-camera, in post-processing, or as a combination of the two.
Take a Class With Dave and Dave
NOTE: This is a direct photo of the cover of my abstract algebra book - the only photoshop I used was to crop it a bit :-) Cool to see a book cover so close, huh? Should be looked at large to get the cool pixel effect!!
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