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Just lines, no curves,
Just light, no shadow,
Just heat, never cold,
Two children, one God,
The Helios's children are here,
on this crossed horizon,
waiting for...
the rain...
This 21 shot panoramic sequence composite was shot at Peek 'n Peak Resort in Clymer, NY. The Math Curve in the title is important since at each point the zipline would make a v-shape which made for the connections between composites to contain cusps instead of smooth flow between the images. To remedy this, I used mathematical methods to fit a high degree polynomial function and then blended that curve into the composite. There are many mathematical questions that can be discussed using this image as a starting point. If you'd like to know some that quickly came to my mind, just let me know in the comments below.
Detail of Mandelbrot set...mapping of quadratic polynomial equation. Saturation and hue modified in Gimp.
Holly: I made a pie to celebrate National Pi Day.
Briefly Pi or π is a mathematical constant, the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century though it is also sometimes spelled out as "pi" (/paɪ/).
Being an irrational number, π cannot be expressed exactly as a common fraction. Consequently its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed although no proof of this has yet been discovered. Also, π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge.
Also Happy Birthday Albert Einstein! I would like to point out to Kass that this pecan pie has lots of eggs in it. Mac was here poking around earlier. Now her and I will have some pie and milk (more protein).
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.
The Silicon Graphics head in my office was my muse. I just finished reading a fascinating summary by Lin & Tegmark of the tie between the power of neural networks / deep learning and the peculiar physics of our universe. The mystery of why they work so well may be resolved by seeing the resonant homology across the information-accumulating substrate of our universe, from the base simplicity of our physics to the constrained nature of the evolved and grown artifacts all around us. The data in our natural world is the product of a hierarchy of iterative algorithms, and the computational simplification embedded within a deep learning network is also a hierarchy of iteration. Since neural networks are symbolic abstractions of how the human cortex works, perhaps it should not be a surprise that the brain has evolved structures that are computationally tuned to tease apart the complexity of our world.
Does anyone know about other explorations into these topics?
Here is a collection of interesting plain text points I extracted from the math in Lin & Tegmark’s article:
"The exceptional simplicity of physics-based functions hinges on properties such as symmetry, locality, compositionality and polynomial log-probability, and we explore how these properties translate into exceptionally simple neural networks approximating both natural phenomena such as images and abstract representations thereof such as drawings. We further argue that when the statistical process generating the data is of a certain hierarchical form prevalent in physics and machine-learning, a deep neural network can be more efficient than a shallow one. Various “no-flattening theorems” show when these efficient deep networks cannot be accurately approximated by shallow ones without efficiency loss."
This last point reminds me of something I wrote in 2006: "Stephen Wolfram’s theory of computational equivalence suggests that simple, formulaic shortcuts for understanding evolution (and neural networks) may never be discovered. We can only run the iterative algorithm forward to see the results, and the various computational steps cannot be skipped. Thus, if we evolve a complex system, it is a black box defined by its interfaces. We cannot easily apply our design intuition to the improvement of its inner workings. We can’t even partition its subsystems without a serious effort at reverse-engineering." — 2006 MIT Tech Review
Back to quotes from the paper:
Neural networks perform a combinatorial swindle, replacing exponentiation by multiplication: if there are say n = 106 inputs taking v = 256 values each, this swindle cuts the number of parameters from v^n to v×n times some constant factor. We will show that this success of this swindle depends fundamentally on physics: although neural networks only work well for an exponentially tiny fraction of all possible inputs, the laws of physics are such that the data sets we care about for machine learning (natural images, sounds, drawings, text, etc.) are also drawn from an exponentially tiny fraction of all imaginable data sets. Moreover, we will see that these two tiny subsets are remarkably similar, enabling deep learning to work well in practice.
Increasing the depth of a neural network can provide polynomial or exponential efficiency gains even though it adds nothing in terms of expressivity.
Both physics and machine learning tend to favor Hamiltonians that are polynomials — indeed, often ones that are sparse, symmetric and low-order.
1. Low polynomial order
For reasons that are still not fully understood, our universe can be accurately described by polynomial Hamiltonians of low order d. At a fundamental level, the Hamiltonian of the standard model of particle physics has d = 4. There are many approximations of this quartic Hamiltonian that are accurate in specific regimes, for example the Maxwell equations governing electromagnetism, the Navier-Stokes equations governing fluid dynamics, the Alv ́en equations governing magnetohydrodynamics and various Ising models governing magnetization — all of these approximations have Hamiltonians that are polynomials in the field variables, of degree d ranging from 2 to 4.
2. Locality
One of the deepest principles of physics is locality: that things directly affect only what is in their immediate vicinity. When physical systems are simulated on a computer by discretizing space onto a rectangular lattice, locality manifests itself by allowing only nearest-neighbor interaction.
3. Symmetry
Whenever the Hamiltonian obeys some symmetry (is invariant under some transformation), the number of independent parameters required to describe it is further reduced. For instance, many probability distributions in both physics and machine learning are invariant under translation and rotation.
Why Deep?
What properties of real-world probability distributions cause efficiency to further improve when networks are made deeper? This question has been extensively studied from a mathematical point of view, but mathematics alone cannot fully answer it, because part of the answer involves physics. We will argue that the answer involves the hierarchical/compositional structure of generative processes together with inability to efficiently “flatten” neural networks reflecting this structure.
A. Hierarchical processes
One of the most striking features of the physical world is its hierarchical structure. Spatially, it is an object hierarchy: elementary particles form atoms which in turn form molecules, cells, organisms, planets, solar systems, galaxies, etc. Causally, complex structures are frequently created through a distinct sequence of simpler steps.
We can write the combined effect of the entire generative process as a matrix product.
If a given data set is generated by a (classical) statistical physics process, it must be described by an equation in the form of [a matrix product], since dynamics in classical physics is fundamentally Markovian: classical equations of motion are always first order differential equations in the Hamiltonian formalism. This technically covers essentially all data of interest in the machine learning community, although the fundamental Markovian nature of the generative process of the data may be an in-efficient description.
Summary
The success of shallow neural networks hinges on symmetry, locality, and polynomial log-probability in data from or inspired by the natural world, which favors sparse low-order polynomial Hamiltonians that can be efficiently approximated. Whereas previous universality theorems guarantee that there exists a neural network that approximates any smooth function to within an error ε, they cannot guarantee that the size of the neural network does not grow to infinity with shrinking ε or that the activation function σ does not become pathological. We show constructively that given a multivariate polynomial and any generic non-linearity, a neural network with a fixed size and a generic smooth activation function can indeed approximate the polynomial highly efficiently.
The success of deep learning depends on the ubiquity of hierarchical and compositional generative processes in physics and other machine-learning applications.
And thanks to Tech Review for the pointer to this article:
Squeezed as much details as I could out of the previous image. Removed light pollution by polynomial modeling and subtracting the interstellar brightness.
5x60s 200mm ISO F2.8@400
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The Bézier Apartments, Old Street, London, England
Well, few days ago I had the plan to go to Brighton and get some LE shots of the piers down there etc but my local trains were heavily delayed so lost a few hours so I decided just a quick visit into London to visit the Bezier Apartments at Old Street. Gosh, wasn't everything against me that day, the clouds were very flat and lacked any definition but gave a long exposure ago.
First I tried a few shops and they just looked rubbish and this shot you see here was totally by accident, I stopped the timer halfway through the timer by accident. It was a bit underexposed but it did show some slight movement in the clouds. With a bit of work in Lightroom and Silver Efex Pro I managed to save it and this was the result. Sadly it does have a little bit of noise in the clouds due to it being slightly underexposed at the start but pushing the noise reduction would only soften the image too much. The end result I am happy with and glad I managed to save something from the day.
I can't find too much about the building but it was designed by architect TP Bennett. The curves are based on the Bezier Curve which is a mathematical calculation which creates this curve.
All in all it was a rubbish day but as photographers I suppose we have to endure the bad days we get to get the good ones that come round, don't we? ;-)
I hope you all having a fantastic weekend!
Photo Details
Sony Alpha SLT-A99 / ISO100 / f/10 / 112s / Hitech ProStop IRND 10 + 6 Stop / Minolta AF 17-35mm F2.8-4 D @ 35mm
Software Used
Lightroom 5
Silver Efex Pro 2
Location Information
A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.
In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs are combinations of linked Bézier curves. Paths are not bound by the limits of rasterized images and are intuitive to modify. Bézier curves are also used in animation as a tool to control motion.
Bézier curves are also used in the time domain, particularly in animation and interface design, e.g., a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or interface designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.
The mathematical basis for Bézier curves — the Bernstein polynomial — has been known since 1912, but its applicability to graphics was understood half a century later. Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies at Renault. The study of these curves was however first developed in 1959 by mathematician Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves, at Citroën, another French automaker.
Quite the perfectionist, Babbage designed this incredibly complicated 5-ton mechanical computer with 8,000 parts in 1822-1849, before the era of standard screws. It would calculate and print tables of solutions to seventh-order polynomials using only addition. Half of the parts are dedicated to the creation of book printing templates to remove all possible sources of human error in the tables, a source of great irritation to Babbage.
While Babbage never saw one built during his lifetime, Nathan Myhrvold commissioned the construction of this one, which is cranking away at the Computer History through the end of this year.
I took an HD video of the beautiful ripples of movement through the soul of the machine.
In optics, defocus is the aberration in which an image is simply out of focus. This aberration is familiar to anyone who has used a camera, videocamera, microscope, telescope, or binoculars. Optically, defocus refers to a translation along the optical axis away from the plane or surface of best focus. In general, defocus reduces the sharpness and contrast of the image. What should be sharp, high-contrast edges in a scene become gradual transitions. Fine detail in the scene is blurred or even becomes invisible. Nearly all image-forming optical devices incorporate some form of focus adjustment to minimize defocus and maximize image quality.
The degree of image blurring for a given amount of focus shift depends inversely on the lens f-number. Low f-numbers, such as f/1.4 to f/2.8, are very sensitive to defocus and have very shallow depths of focus. High f-numbers, in the f/16 to f/32 range, are highly tolerant of defocus, and consequently have large depths of focus. The limiting case in f-number is the pinhole camera, operating at perhaps f/100 to f/1000, in which case all objects are in focus almost regardless of their distance from the pinhole aperture. The penalty for achieving this extreme depth of focus is very dim illumination at the imaging film or sensor, limited resolution due to diffraction, and very long exposure time, which introduces the potential for image degradation due to motion blur.
The amount of allowable defocus is related to the resolution of the imaging medium. A lower-resolution imaging chip or film is more tolerant of defocus and other aberrations. To take full advantage of a higher resolution medium, defocus and other aberrations must be minimized.
Defocus is modeled in Zernike polynomial format as , where is the defocus coefficient in wavelengths of light. This corresponds to the parabola-shaped optical path difference between two spherical wavefronts that are tangent at their vertices and have different radii of curvature.
For some applications, such as phase contrast electron microscopy, defocused images can contain useful information. Multiple images recorded with various values of defocus can be used to examine how the intensity of the electron wave varies in three dimensional space, and from this information the phase of the wave can be inferred. This is the basis of non-interferometric phase retrieval. Examples of phase retrieval algorithms that utilise defocused images include the Gerchberg–Saxton algorithm and various methods based on the transport of intensity equation.
Each tiny disk corresponds to a complex root of a polynomial with bounded coefficients. In particular the polynomial coefficients come from the set {-1,0,1}. All polynomials with degree <= 11, with a 1 for the constant term are used. As such each polynomial corresponds to a unique integer obtained by its evaluation at 3. The absolute value of this integer is used to determine the size and color of each disk. If I recall correctly there are about a half a million dots.
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).
Archimedes Liu Hui Zu Chongzhi Madhava of Sangamagrama William Jones John Machin John Wrench Ludolph van Ceulen Aryabhata
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v t e
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.
© Saúl Tuñón Loureda
El Museo Británico (en inglés: British Museum) es un museo de la ciudad de Londres, Reino Unido, uno de los museos más importantes y visitados del mundo. Sus colecciones abarcan campos diversos del saber humano, como la historia, la arqueología, la etnografía y el arte.
El museo fue una de las primeras instituciones de este tipo en Europa. Custodia más de siete millones de objetos de todos los continentes, muchos de los cuales se encuentran almacenados para su estudio y restauración, o guardados por falta de espacio para exhibirlos. Cuenta con la mayor sala de lectura de la Biblioteca Británica, biblioteca que aunque ahora tiene sede propia, hasta el año 1973 también formaba parte del museo, al igual que el Museo de Historia Natural de Londres, que cambió a sede propia en el año 1963.
La sección del Antiguo Egipto es la más importante del mundo después de la del Museo Egipcio de El Cairo. La entrada al museo y a muchos de los servicios que ofrece —como el de la sala de lectura— es libre y gratuita, a excepción de algunas exposiciones temporales.
es.wikipedia.org/wiki/Museo_Brit%C3%A1nico
The British Museum is a museum dedicated to human history, art, and culture, located in the Bloomsbury area of London. Its permanent collection, numbering some 8 million works,[3] is among the largest and most comprehensive in existence[3] and originates from all continents, illustrating and documenting the story of human culture from its beginnings to the present.[a]
The British Museum was established in 1753, largely based on the collections of the physician and scientist Sir Hans Sloane. The museum first opened to the public on 15 January 1759, in Montagu House in Bloomsbury, on the site of the current museum building. Its expansion over the following two and a half centuries was largely a result of an expanding British colonial footprint and has resulted in the creation of several branch institutions, the first being the British Museum (Natural History) in South Kensington in 1881. Some objects in the collection, most notably the Elgin Marbles from the Parthenon, are the objects of controversy and of calls for restitution to their countries of origin.
Until 1997, when the British Library (previously centred on the Round Reading Room) moved to a new site, the British Museum housed both a national museum of antiquities and a national library in the same building. The museum is a non-departmental public body sponsored by the Department for Culture, Media and Sport, and as with all other national museums in the United Kingdom it charges no admission fee, except for loan exhibitions. Neil MacGregor became director of the museum in August 2002, succeeding Robert G. W. Anderson. In April 2015, MacGregor announced that he would step-down as Director on 15 December. On 29 September 2015, the Board of Trustees confirmed Hartwig Fischer, who will assume his post in Spring 2016, as his successor.
Title (object)
Hoa Hakananai'a ('lost or stolen friend')
Moai (ancestor figure)
Description
Ancestor figure 'moai', called Hoa Hakananai'a (hidden or stolen friend) made of basalt. Images relating to the bird man religion (tangata manu); birds, vulvas, dance paddles in the form of stylizes human figure, a ring and a girdle design are carved in relief on the back of the figure's head and body.
en.wikipedia.org/wiki/British_Museum
Date
1000 -1200 (approx)
Production place
Made in: Rano Kao (likely)
(Oceania,Polynesia,Easter Island,Rano Kao)
Findspot
Found/Acquired: Orongo, ceremonial centre
(Oceania,Polynesia,Easter Island,Orongo)
Materials
stone (missing (eyes sockets)) term details
coral (missing (eyes sockets))
basalt
Technique
painted (previously) term details
inlaid (previously)
carved
Dimensions
Height: 2.42 metres
Width: 96 centimetres
Diameter: 47 centimetres
Curator's comments
Ethnography Department Temporary Register, 1861-1921
It is understood that large stone sculptures or moai were made on Rapa Nui between AD 1100 and 1600. The size and complexity of the moai increased over time, and it is believed that Hoa Hakananai'a dates to around AD 1200. It is one of only fourteen moai made from basalt, the rest are carved from the island’s softer volcanic tuff.
This statue would have originally stood on a specially-built platform on the sacred site of Orongo. It would have stood with giant stone companions, their backs to the sea, keeping watch over the island.
Its eyes sockets were originally inlaid with red stone and coral and the sculpture was painted with red and white designs, which were washed off when it was rafted to the ship, to be taken to Europe in 1869. Over a few hundred years the inhabitants of this remote island quarried, carved and erected around 887 moai.
This sculpture bears witness to the loss of confidence in the efficacy of the ancestors after the deforestation and ecological collapse, and most recently a theory concerning the introduction of rats, which may have ultimately led to famine and conflict.
Around AD 1500 the practice of constructing moai peaked, and from around AD 1600 statues began to be toppled, sporadically. The island’s fragile ecosystem had been pushed beyond what was sustainable. Over time only sea birds remained, nesting on safer offshore rocks and islands. As these changes occurred, so too did the Rapanui religion alter – to the birdman religion.
A project to record and analyse the statue's carvings took place on 15th February 2012 in the Wellcome Trust Gallery. The techniques used were photogrammetry and polynomial texture mapping (PTM). It was conducted by Mike Pitts, Graeme Earl, James Miles and Hembo Pagi in collaboration with Southampton University. This is the first Easter Island statue to be so fully described.
For more detailed information see: Pitts, M., Miles, J., Pagi, H. and Earl, G., The story of Hoa Hakananai'a in British Archaeology: May/June 2013 and visit mikepitts.wordpress.com/easter-island-rapa-nui/
See bibliography:
Van Tilburg, J.A.,1992, 'H.M.S. Topaze on Easter Island: Hoa Hakananai'a and five other museum sculptures in archaeological context.' London: British Museum Press, Occasional paper 73.
Van Tilburg, J.A., 1994, 'Easter Island, Archaeology, Ecology and Culture'. London: British Museum Press.
Van Tilburg, J.A., 2004, 'Hoa Hakananai'a'. London: British Museum Press.
Van Tilburg, J.A., 2006, 'Remote Possibilities: Hoa Hakananai'a and HMS Topaze on Rapa Nui'. London: British Museum Press.
Hoa Hakananai'a is estimated to weigh approx. 4.2 metric tonnes (including the plinth it stands on, 5 tonnes) - 2013.
More
Bibliography
MacGregor 2010 70 bibliographic details
Hooper 2006 p.20, fig. 6 bibliographic details
Brunt et al, 2012 p.253 bibliographic details
Newell 2011 p.66/67 bibliographic details
Location
On display: G24/od
Exhibition history
Exhibited:
2010 Sept-Dec, London, BM History of the World 100 objects
2010-2011, London, BM/BBC, 'A History of the World in 100 Objects'
Subjects
nationality/peoples term details
bird
anthropomorphism term details
Acquisition name
Collected by: Cdre Richard Ashmore Powell
Donated by: Lords of Admiralty biography
Donated by: Queen Victoria biography
Acquisition date
1869
Acquisition notes
Collected during the HMS Topaze expedition to Rapa Nui (captained by Powell) in 1868 and presented to Queen Victoria by the Lords of the Admiralty. She then gifted it to the British Museum in 1869.
Department
Africa, Oceania & the Americas
www.britishmuseum.org/research/collection_online/collecti...
Aphex Twin
Book :
Diane Arbus
Schirmer / Mosel
2017
CD :
Aphex Twin
The Classics
R & S Records
RS95035
Sounds . Richard D. James
Design . Paul Nicholson
iMusic :
The Go - Betweens
The Clarke Sisters
Beggars Banquet
BEGA81
Polynomial - GMA ...
You don't get to the top of this hill by adding complex fractions or finding the leading coefficient of the polynomial.
1:64 GreenLight Collectibles:
1977 Dodge Macho
Ramcharger 4x4
Four By Four Stripe Kit
All-Terrain Series 10
Olympus OM-D E-M5 Mark II
Olympus M.14-42mm F3.5-5.6 II R
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