Geometrical Figures XVII: Bell Curves

"A bell curve shows the 'spread' or variance in our knowledge or certainty. The wider the bell the less we know. An infinitely wide bell is a flat line. Then we know nothing. The value of the quantity, position, or speed could lie anywhere on the axis. An infinitely narrow bell is a spike that is infinitely tall. Then we have complete knowledge of the value of the quantity. The uncertainty principle says that as one bell curve gets wider the other gets thinner. As one curve peaks the other spreads. So if the position bell curve becomes a spike and we have total knowledge of position, then the speed bell curve goes flat and we have total uncertainty (infinite variance) of speed." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"Every network has its own fitness distribution, which tells us how similar or different the nodes in the network are. In networks where most of the nodes have comparable fitness, the distribution follows a narrowly peaked bell curve. In other networks, the range of fitnesses is very wide such that a few nodes are much more fit than most others. […] the mathematical tools developed decades earlier to describe quantum gases enabled us to see that, independent of the nature of links and nodes, a network's behavior and topology are determined by the shape of its fitness distribution. But even though each system, from the Web to Holywood, has a unique fitness distribution, Bianconi's calculation indicated that in terms of topology all networks fall into one of only two possible categories. In most networks the competition does not have an easily noticeable impact on the network's topology. In some networks, however, the winner takes all the links, a clear signature of Bose-Einstein condensation." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge-nature's unmistakable sign that chaos is departing in favor of order. The theory of phase transitions told us loud and clear that the road from disorder to order is maintained by the powerful forces of self-organization and is paved by power laws. It told us that power laws are not just another way of characterizing a system's behavior. They are the patent signatures of self-organization in complex systems." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Bell curves don't differ that much in their bells. They differ in their tails. The tails describe how frequently rare events occur. They describe whether rare events really are so rare. This leads to the saying that the devil is in the tails." (Bart Kosko, "Noise", 2006)

"Many scientists who work not just with noise but with probability make a common mistake: They assume that a bell curve is automatically Gauss's bell curve. Empirical tests with real data can often show that such an assumption is false. The result can be a noise model that grossly misrepresents the real noise pattern. It also favors a limited view of what counts as normal versus non-normal or abnormal behavior. This assumption is especially troubling when applied to human behavior. It can also lead one to dismiss extreme data as error when in fact the data is part of a pattern." (Bart Kosko, "Noise", 2006)

"The central limit theorem differs from laws of large numbers because random variables vary and so they differ from constants such as population means. The central limit theorem says that certain independent random effects converge not to a constant population value such as the mean rate of unemployment but rather they converge to a random variable that has its own Gaussian bell-curve description." (Bart Kosko, "Noise", 2006)

"The flaw in the classical thinking is the assumption that variance equals dispersion. Variance tends to exaggerate outlying data because it squares the distance between the data and their mean. This mathematical artifact gives too much weight to rotten apples. It can also result in an infinite value in the face of impulsive data or noise. [...] Yet dispersion remains an elusive concept. It refers to the width of a probability bell curve in the special but important case of a bell curve. But most probability curves don't have a bell shape. And its relation to a bell curve's width is not exact in general. We know in general only that the dispersion increases as the bell gets wider. A single number controls the dispersion for stable bell curves and in-deed for all stable probability curves - but not all bell curves are stable curves."  (Bart Kosko, "Noise", 2006)

"Before calculating a confidence interval for a mean, first check that one of the situations just described holds. To determine whether the data are bell-shaped or skewed, and to check for outliers, plot the data using a histogram, dotplot, or stemplot. A boxplot can reveal outliers and will sometimes reveal skewness, but it cannot be used to determine the shape otherwise. The sample mean and median can also be compared to each other. Differences between the mean and the median usually occur if the data are skewed - that is, are much more spread out in one direction than in the other." (Jessica M Utts & Robert F Heckard, "Mind on Statistics", 2007)

"Symmetry and skewness can be judged, but boxplots are not entirely useful for judging shape. It is not possible to use a boxplot to judge whether or not a dataset is bell-shaped, nor is it possible to judge whether or not a dataset may be bimodal." (Jessica M Utts & Robert F Heckard, "Mind on Statistics", 2007)

"With time series though, there is absolutely no substitute for plotting. The pertinent pattern might end up being a sharp spike followed by a gentle taper down. Or, maybe there are weird plateaus. There could be noisy spikes that have to be filtered out. A good way to look at it is this: means and standard deviations are based on the naïve assumption that data follows pretty bell curves, but there is no corresponding 'default' assumption for time series data (at least, not one that works well with any frequency), so you always have to look at the data to get a sense of what’s normal. [...] Along the lines of figuring out what patterns to expect, when you are exploring time series data, it is immensely useful to be able to zoom in and out." (Field Cady, "The Data Science Handbook", 2017)

"Skewed data means data that is shifted in one direction or the other. Skewness can cause machine learning models to underperform. Many machine learning models assume normally distributed data or data structures to follow the Gaussian structure. Any deviation from the assumed Gaussian structure, which is the popular bell curve, can affect model performance. A very effective area where we can apply feature engineering is by looking at the skewness of data and then correcting the skewness through normalization of the data." (Anthony So et al, "The Data Science Workshop" 2nd Ed., 2020)

Figurative Figures III: Curves

“Truth is on a curve whose asymptote our spirit follows eternally." (Léo Errera,"Recueil d'Œuvres de Léo Errera: Botanique Générale", 1908)

"Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line [...]" (Vladimir I Lenin, "On the Question of Dialectics", 1915)

No revolution, no heresy is comfortable or easy. For it is a leap, it is a break in the smooth evolutionary curve, and a break is a wound, a pain. But the wound is necessary; most of mankind suffers from hereditary sleeping sickness, and victims of this sickness (entropy) must not be allowed to sleep, or it will be their final sleep, death. (Yevgeny Zamiatin, "On Literature, Revolution, Entropy, and Other Matters", 1923)

"Is evolution a theory, a system or a hypothesis? It is much more: it is a general condition to which all theories, all hypotheses, all systems must bow and which they must satisfy henceforth if they are to be thinkable and true. Evolution is a light illuminating all facts, a curve that all lines must follow." (Pierre T de Chardin, "The Phenomenon of Man", 1955)

The problems are the ones that we have always known. The little gods are still with us, under different names. There is conformity: of technique, leading to repetition; of language, encouraging if not imposing conformity of thought. There is popularity: it is so easy to ride along on an already surging tide; to plant more seed in an already well-ploughed field; so hard to drive a new furrow into stony ground. There is laxness: the disregard of small errors, of deviations, of the unexpected response; the easy worship of the smooth curve. There is also fear: the fear of speculation; the overprotective fear of being wrong. We are forgetful of the curious and wayward dialectic of science, whereby a well-constructed theory even if it is wrong, can bring a signal advance. (Dickinson W Richards, Transactions of the Association of American Physicians Vol. 75, 1962)

"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring Mathematics All Around You", 1986)

"The metaphor never goes very far, anymore than a curve can long be confused with its tangent." (Henri Bergson,"A World of Ideas", 1989)

"History too has an inertia. In the four dimensions of spacetime, particles (or events) have directionality; mathematicians, trying to show this, draw what they call 'world lines' on graphs. In human affairs, individual world lines form a thick tangle, curling out of the darkness of prehistory and stretching through time: a cable the size of Earth itself, spiraling round the sun on a long curved course. That cable of tangled world lines is history. Seeing where it has been, it is clear where it is going - it is a matter of simple extrapolation." (Kim S Robinson, "Red Mars", 1992)

Curved Spaces

"The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon [...] when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms." (Jean-Baptiste-Joseph Fourier, "Théorie Analytique de la Chaleur", 1822)

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870)

[...] the time stream is curved helically in some higher dimension. In your case, a still further distortion brought two points of the coil into contact, and a sort of short circuit threw you into the higher curve. (Robert H Wilson, "A Flight Into Time", Wonder Stories, 1931)

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947) 

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system. (James Gleick, "Chaos: Making a New Science", 1987)

"Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest (or longest) path between two nearby points." (Stephen Hawking, "A Brief History of Time", 1988)

"Nonlinear systems (the graph of at least one relationship displays some curved feature) are notoriously more difficult to comprehend than linear systems, that is, they are more complex. Consequently they are also more difficult to control. This is exemplified by the volumes of elegant mathematics that have been developed in the search for optimal control of linear systems. (Robert L Flood & Ewart R Carson, "Dealing with Complexity: An introduction to the theory and application of systems", 1988)

"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system." (James Gleick, "Chaos: Making a New Science", 1987)

"String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time." (John D. Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Spacetime tells matter how to move; matter tells spacetime how to curve." (John A Wheeler, "Geons, Black Holes and Quantum Foam: A Life in Physics" , 1998)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers (and physicists) cannot do without them. Thus there is no doubt that mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)