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This page is part of the Mathematics Portal |

This is the collection of pictures that are being randomly selected for display on Portal:Mathematics. (This system replaces the previous " Featured picture" system that was in use until March 2014.)

**To add a new image:**

- Edit any existing subpage in the list below (link in upper-left corner of box) and copy its wikicode.
- Select the first non-existing subpage listed below (shown as a "redlink") to start editing that subpage, and paste the wikicode you just copied.
- If there is no "redlink" at the bottom of the list, edit
*this*page and look for the code`{{numbered subpages|max=`

. Increase*nn*}}*nn*by 2 (so there will still be a redlink after you create one new supage) and save the page.

- If there is no "redlink" at the bottom of the list, edit
- Change all the relevant template parameters, as necessary.
- Note that the value of the
`caption`

parameter is used as the image's " alt text" and so should describe the*appearance*of the image as if explaining it to someone who cannot see it. You need not use a complete, grammatically correct sentence for this (e.g., "animation of*blah*" can be used instead of "This is an animation of*blah*."). - The
`text`

parameter is displayed as a paragraph of text below the image and should use complete, grammatically correct sentences to explain the*meaning*of the image and its significance from a mathematical perspective. - If the code you copied in step 1 used the
`size`

parameter to specify the image size, try removing that line to see whether the image looks acceptable at the default size. - Finally, note that we do not use the
`gallery`

,`location`

or`archive`

parameters of the {{ Selected picture}} template. Also, please do not change the`page = picture`

or`framecolor = transparent`

lines.)

- Note that the value of the
- Save the new subpage and check that it's being displayed correctly in the list below. (You may have to purge the cache to see the changes.)
- If the default size is not acceptable (for example, if a
raster image is being displayed
*larger*than its actual size or if a GIF animation is showing up as a static image), the`size`

parameter can be used to set the width (in pixels) of the image.

- If the default size is not acceptable (for example, if a
raster image is being displayed
- Edit
Portal:Mathematics and locate the template call that randomly chooses the pictures for display:
`{{Random portal component|max=`

*nn*|subpage=Selected picture|header=Selected picture}}

- Increase the number
to reflect the new count of selected pictures, and save the page.*nn* - And you're done.

**To find good images** to add here, see any of the following image collections:

- Portal:Mathematics/Featured picture archive (there are still a few good images that haven't been ported from the old system)
- Wikipedia:WikiProject Mathematics#Featured images
- Wikipedia:Featured pictures/Sciences/Mathematics
- Wikipedia:Featured pictures/Diagrams, drawings, and maps/Diagrams
- Commons:Category:Featured diagrams
- Commons:Category:Mathematics

You can also simply look through the images currently being displayed in our mathematics articles:

- Wikipedia:WikiProject Mathematics/Recognized content
- Wikipedia:Featured articles#Mathematics (consider also other sections on related fields, like economics, engineering, physics, astronomy, etc.)
- Wikipedia:Good articles/Mathematics
- Lists of mathematics topics
- Category:Mathematics

Try to select new images from areas of mathematics that are not already represented in the list below.

Portal:Mathematics/Selected picture/1

Portal:Mathematics/Selected picture/2

Credit:
TakuyaMurata (uploader)

This is a hand-drawn graph of the
absolute value (or modulus) of the **
gamma function** on the
complex plane, as published in the 1909 book *Tables of Higher Functions*, by Eugene Jahnke and Fritz Emde. Such three-dimensional graphs of complicated
functions were rare before the advent of high-resolution
computer graphics (even today,
tables of values are used in many contexts to look up function values instead of consulting graphs directly). Published even before applications for the complex gamma function were discovered in theoretical physics in the 1930s, Jahnke and Emde's graph "acquired an almost iconic status", according to physicist
Michael Berry. See
a similar computer-generated image for comparison. When restricted to positive integers, the gamma function generates the
factorials through the relation Γ(*n*) = (*n* − 1)!, which is the product of all positive integers from *n* − 1 down to 1 (0! is defined to be equal to 1). For
real and
complex numbers, the function is defined by the
improper integral . This integral
diverges when *t* is a negative integer, which is causing the spikes in the left half of the graph (these are
simple poles of the function, where its values increase to
infinity, analogous to
asymptotes in two-dimensional graphs). The gamma function has applications in
quantum physics,
astrophysics, and
fluid dynamics, as well as in
number theory and
probability.

Portal:Mathematics/Selected picture/3

Credit:
Wikimol

The **
Lorenz attractor** is an iconic example of a
strange attractor in
chaos theory. This three-dimensional
fractal structure, resembling a
butterfly or
figure eight, reflects the long-term behavior of solutions to the
Lorenz system, a set of three
differential equations used by mathematician and meteorologist
Edward N. Lorenz as a simple description of fluid circulation in a shallow layer (of liquid or gas) uniformly heated from below and cooled from above. To be more specific, the figure is set in a three-dimensional coordinate system whose axes measure the rate of convection in the layer (*x*), the horizontal temperature variation (*y*), and the vertical temperature variation (*z*). As these quantities change over time, a path is traced out within the coordinate system reflecting a particular solution to the differential equations. Lorenz's analysis revealed that while all solutions are completely
deterministic, some choices of input parameters and initial conditions result in solutions showing complex, non-repeating patterns that are highly dependent on the exact values chosen. As stated by Lorenz in his 1963 paper *Deterministic Nonperiodic Flow*: "Two states differing by imperceptible amounts may eventually evolve into two considerably different states". He later coined the term "
butterfly effect" to describe the phenomenon. One implication is that computing such chaotic solutions to the Lorenz system (i.e., with a computer program) to arbitrary precision is not possible, as any real-world computer will have a limitation on the precision with which it can represent numerical values. The particular solution plotted in this animation is based on the parameter values used by Lorenz (*σ* = 10, *ρ* = 28, and *β* = 8/3, constants reflecting certain physical attributes of the fluid). Note that the animation repeatedly shows one solution plotted over a specific period of time; as previously mentioned, the true solution never exactly retraces itself. Not all solutions are chaotic, however. Some choices of parameter values result in solutions that tend toward
equilibrium at a fixed point (as seen, for example, in
this image). Initially developed to describe atmospheric convection, the Lorenz equations also arise in simplified models for
lasers,
electrical generators and motors, and
chemical reactions.

Portal:Mathematics/Selected picture/4

Here a
polyhedron called a **
truncated icosahedron** (left) is compared to the classic
Adidas Telstar–style
football (or soccer ball). The familiar 32-panel ball design, consisting of 12 black
pentagonal and 20 white
hexagonal panels, was first introduced by the Danish manufacturer
Select Sport, based loosely on the
geodesic dome designs of
Buckminster Fuller; it was popularized by the selection of the Adidas Telstar as the official match ball of the
1970 FIFA World Cup. The polyhedron is also the shape of the
Buckminsterfullerene (or "Buckyball") carbon molecule initially predicted theoretically in the late 1960s and first generated in the laboratory in 1985. Like all
polyhedra, the vertices (corner points), edges (lines between these points), and faces (flat surfaces bounded by the lines) of this solid obey the
Euler characteristic, *V* − *E* + *F* = 2 (here, 60 − 90 + 32 = 2). The
icosahedron from which this solid is obtained by
truncating (or "cutting off") each vertex (replacing each by a pentagonal face), has 12 vertices, 30 edges, and 20 faces; it is one of the five regular solids, or
Platonic solids—named after
Plato, whose
school of philosophy in ancient Greece held that the
classical elements (earth, water, air, fire, and a fifth element called
aether) were associated with these regular solids. The fifth element was known in
Latin as the "quintessence", a hypothesized uncorruptible material (in contrast to the other four terrestrial elements) filling the heavens and responsible for celestial phenomena. That such idealized mathematical shapes as polyhedra actually occur in nature (e.g., in
crystals and other
molecular structures) was discovered by naturalists and physicists in the 19th and 20th centuries, largely independently of the ancient philosophies.

Portal:Mathematics/Selected picture/5

Credit:
Ilmari Karonen

The **
knight's tour** is a
mathematical chess problem in which the piece called the
knight is to visit each square on an otherwise empty
chess board exactly once, using only legal moves. It is a special case of the more general
Hamiltonian path problem in
graph theory. (A closely related non-Hamiltonian problem is that of the
longest uncrossed knight's path.) The tour is called *closed* if the knight ends on a square from which it may legally move to its starting square (thereby
forming an endless cycle), and *open* if not. The tour shown in this animation is open (see also a
static image of the completed tour). On a standard 8 × 8 board there are 26,534,728,821,064 possible closed tours and 39,183,656,341,959,810 open tours (counting separately any tours that are
equivalent by rotation, reflection, or reversing the direction of travel). Although the earliest known solutions to the knight's tour problem date back to the 9th century
CE, the first general procedure for completing the knight's tour was
Warnsdorff's rule, first described in 1823. The knight's tour was one of many
chess puzzles solved by
The Turk, a fake chess-playing machine exhibited as an
automaton from 1770 to 1854, and exposed in the early 1820s as an elaborate hoax. True
chess-playing automatons (i.e., computer programs) appeared in the 1950s, and by 1988 had become sufficiently advanced to win a match against a
grandmaster; in 1997,
Deep Blue famously became the first computer system to defeat a reigning world champion (
Garry Kasparov) in a match under standard tournament time controls. Despite these advances, there is still debate as to whether
chess will ever be "solved" as a computer problem (meaning an algorithm will be developed that can never lose a chess match). According to
Zermelo's theorem, such an algorithm does exist.

Portal:Mathematics/Selected picture/6

This spiral diagram represents all **
ordinal numbers** less than *ω*^{ω}. The first (outermost) turn of the spiral represents the finite ordinal numbers, which are the regular
counting numbers starting with
zero. As the spiral completes its first turn (at the top of the diagram), the ordinal numbers approach
infinity, or more precisely ω, the first
transfinite ordinal number (identified with the set of all counting numbers, a "
countably infinite" set, the
cardinality of which corresponds to the first transfinite
cardinal number, called ℵ_{0}). The ordinal numbers continue from this point in the second turn of the spiral with *ω* + 1, *ω* + 2, and so forth. (A special
ordinal arithmetic is defined to give meaning to these expressions, since the + symbol here does not represent the addition of two
real numbers.) Halfway through the second turn of the spiral (at the bottom) the numbers approach *ω* + *ω*, or *ω* · 2. The ordinal numbers continue with *ω* · 2 + 1 through *ω* · 2 + *ω* = *ω* · 3 (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through *ω* · 4, and so forth, up to *ω* · *ω* = *ω*^{2} at the top. (As with addition, the multiplication and exponentiation operations have definitions that work with transfinite numbers.) The ordinals continue in the third turn of the spiral with *ω*^{2} + 1 through *ω*^{2} + *ω*, then through *ω*^{2} + *ω*^{2} = *ω*^{2} · 2, up to *ω*^{2} · *ω* = *ω*^{3} at the top of the third turn. Continuing in this way, the ordinals increase by one power of ω for each turn of the spiral, approaching *ω*^{ω} in the middle of the diagram, as the spiral makes a countably infinite number of turns. This process can actually continue (not shown in this diagram) through and , and so on, approaching the
first epsilon number, *ε*_{0}. Each of these ordinals is still countable, and therefore equal in cardinality to ω. After uncountably many of these transfinite ordinals, the
first uncountable ordinal is reached, corresponding to only the second infinite cardinal, . The identification of this larger cardinality with the
cardinality of the set of real numbers can neither be proved nor disproved within the standard version of
axiomatic set theory called
Zermelo–Fraenkel set theory, whether or not one also assumes the
axiom of choice.

Portal:Mathematics/Selected picture/7

Credit:
Lucas V. Barbosa

A **
line integral** is an
integral where the
function to be integrated, be it a
scalar field as here or a
vector field, is evaluated along a
curve. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly
arc length or, for a vector field, the
scalar product of the vector field with a
differential vector in the curve). A
detailed explanation of the animation is available. The key insight is that line integrals may be reduced to simpler
definite integrals. (See also
a similar animation illustrating a line integral of a vector field.) Many formulas in elementary
physics (for example, *W* = **F** · **s** to find the
work done by a constant force **F** in moving an object through a displacement **s**) have line integral versions that work for non-constant quantities (for example, *W* = ∫_{C} **F** · *d***s** to find the work done in moving an object along a curve *C* within a continuously varying gravitational or electric
field **F**). A higher-dimensional analog of a line integral is a
surface integral, where the (double) integral is taken over a two-dimensional surface instead of along a one-dimensional curve. Surface integrals can also be thought of as generalizations of
multiple integrals. All of these can be seen as special cases of integrating a
differential form, a viewpoint which allows
multivariable calculus to be done independently of the choice of coordinate system. While the
elementary notions upon which integration is based date back centuries before
Newton and Leibniz independently invented calculus, line and surface integrals were formalized in the 18th and 19th centuries as the subject was placed on a rigorous mathematical foundation. The modern notion of differential forms, used extensively in
differential geometry and
quantum physics, was pioneered by
Élie Cartan in the late 19th century.

Portal:Mathematics/Selected picture/8

This image illustrates a failed attempt to comb the "hair" on a ball flat, leaving a tuft sticking out at each pole. The **
hairy ball theorem** of
algebraic topology states that whenever one attempts to comb a hairy ball, there will always be at least one point on the ball at which a tuft of hair sticks out. More precisely, it states that there is no nonvanishing
continuous tangent-
vector field on an even-dimensional
*n*‑sphere (an ordinary sphere in three-dimensional space is known as a "2-sphere"). This is not true of certain other three-dimensional shapes, such as a
torus (doughnut shape) which
*can* be combed flat. The theorem was first stated by
Henri Poincaré in the late 19th century and proved in 1912 by
L. E. J. Brouwer. If one idealizes the wind in the Earth's atmosphere as a tangent-vector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a
cyclone somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not
meteorologically sound. (What *is* true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in
computer modeling (including
video game design), in which a common problem is to compute a non-zero
3-D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task.

Portal:Mathematics/Selected picture/9

Credit:
Phil Tregoning

A **
Bézier curve** is a
parametric curve important in
computer graphics and related fields.
Widely publicized in 1962 by the
French engineer
Pierre Bézier, who used them to design
automobile bodies, the curves were first developed in 1959 by
Paul de Casteljau using
de Casteljau's algorithm. In this animation, a
quartic Bézier curve is constructed using *control points* P_{0} through P_{4}. The green line segments join points moving at a constant rate from one control point to the next; the *parameter* t shows the progress over time. Meanwhile, the blue line segments join points moving in a similar manner along the green segments, and the
magenta line segment points along the blue segments. Finally, the black point moves at a constant rate along the magenta line segment, tracing out the final curve in red. The curve is a
fourth-degree function of its parameter.
Quadratic and
cubic Bézier curves are most common since higher-
degree curves are more computationally costly to evaluate. When more complex shapes are needed, lower-order Bézier curves are patched together. For example, modern
computer fonts use
Bézier splines composed of quadratic or cubic Bézier curves to create scalable
typefaces. The curves are also used in
computer animation and
video games to plot smooth paths of motion. Approximate Bézier curves can be generated in the "real world" using
string art.

Portal:Mathematics/Selected picture/10

Credit:
Steven G. Johnson (original version)

This is a graphical construction of the various **
trigonometric functions** from a
unit circle centered at the
origin, O, and two points, A and D, on the circle separated by a central angle θ. The triangle AOC has side lengths
cos *θ* (OC, the side *adjacent* to the angle θ) and
sin *θ* (AC, the side *opposite* the angle), and a
hypotenuse of length 1 (because the circle has
unit
radius). When the
tangent line AE to the circle at point A is drawn to meet the extension of OD beyond the limits of the circle, the triangle formed, AOE, contains sides of length
tan *θ* (AE) and
sec *θ* (OE). When the tangent line is extended in the other direction to meet the line OF drawn perpendicular to OC, the triangle formed, AOF, has sides of length
cot *θ* (AF) and
csc *θ* (OF). In addition to these common trigonometric functions, the diagram also includes some functions that have fallen into disuse: the
chord (AD),
versine (CD),
exsecant (DE),
coversine (GH), and
excosecant (FH). First used in the early
Middle Ages by
Indian and
Islamic mathematicians to solve simple geometrical problems (e.g.,
solving triangles), the trigonometric functions today are used in sophisticated two- and three-dimensional computer modeling (especially when
rotating modeled objects), as well as in the study of
sound and other
mechanical waves,
light (electromagnetic waves), and
electrical networks.

Portal:Mathematics/Selected picture/11

Credit:
BenFrantzDale

A **
Lorenz curve** shows the distribution of
income in a population by plotting the percentage y of total income that is earned by the bottom x percent of households (or individuals). Developed by economist
Max O. Lorenz in 1905 to describe
income inequality, the curve is typically plotted with a diagonal line (reflecting a hypothetical "equal" distribution of incomes) for comparison. This leads naturally to a derived quantity called the
Gini coefficient, first published in 1912 by
Corrado Gini, which is the ratio of the area between the diagonal line and the curve (area A in this graph) to the area under the diagonal line (the sum of A and B); higher Gini coefficients reflect more income inequality. Lorenz's curve is a special kind of
cumulative distribution function used to characterize quantities that follow a
Pareto distribution, a type of
power law. More specifically, it can be used to illustrate the
Pareto principle, a
rule of thumb stating that roughly 80% of the identified "effects" in a given phenomenon under study will come from 20% of the "causes" (in the first decade of the 20th century
Vilfredo Pareto showed that 80% of the land in Italy was owned by 20% of the population). As this so-called "80–20 rule" implies a specific level of inequality (i.e., a specific power law), more or less extreme cases are possible. For example,
in the United States in the first half of the 2010s, 95% of the financial
wealth was held by the top 20% of wealthiest households (in 2010), the top 1% of individuals held approximately 40% of the wealth (2012), and the top 1% of
income earners received approximately 20% of the pre-tax income (2013). Observations such as these have brought income and wealth inequality into
popular consciousness and have given rise to various slogans about
"the 1%" versus "the 99%".

Portal:Mathematics/Selected picture/12

Credit:
Sam Derbyshire

A **
hypotrochoid** is a curve traced out by a point "attached" to a smaller circle rolling around inside a fixed larger circle. In this example, the hypotrochoid is the red curve that is traced out by the red point 5 units from the center of the black circle of radius 3 as it rolls around inside the blue circle of radius 5. A special case is a hypotrochoid with the inner circle exactly one-half the radius of the outer circle, resulting in an
ellipse (see
an animation showing this). Mathematical analysis of closely-related curves called
hypocycloids lead to special
Lie groups. Both hypotrochoids and
epitrochoids (where the moving circle rolls around on the outside of the fixed circle) can be created using the
Spirograph drawing toy. These curves have applications in the "real world" in
epicyclic and hypocycloidal gearing, which were used in
World War II in the construction of portable
radar gear and may be used today in
3D printing.

Portal:Mathematics/Selected picture/13

This is a graph of a portion of the complex-valued **
Riemann zeta function** along the *critical line* (the set of
complex numbers having real part equal to 1/2). More specifically, it is a graph of Im ζ(1/2 + *it*) versus Re ζ(1/2 + *it*) (the imaginary part vs. the real part) for values of the real variable t running from 0 to 34 (the curve starts at its leftmost point, with real part approximately −1.46 and imaginary part 0). The first five
zeros along the critical line are visible in this graph as the five times the curve passes through the origin (which occur at t
≈ 14.13, 21.02, 25.01, 30.42, and 32.93 — for a different perspective, see
a graph of the real and imaginary parts of this function plotted separately over a wider range of values). In 1914,
G. H. Hardy proved that ζ(1/2 + *it*) has infinitely many zeros. According to the
Riemann hypothesis, zeros of this form constitute the only
non-trivial zeros of the full zeta function, ζ(*s*), where s varies over all complex numbers. Riemann's zeta function grew out of
Leonhard Euler's study of real-valued
infinite series in the early 18th century. In a famous 1859 paper called "
On the Number of Primes Less Than a Given Magnitude",
Bernhard Riemann extended Euler's results to the complex plane and established a relation between the zeros of his zeta function and
the distribution of prime numbers. The paper also contained the previously mentioned Riemann hypothesis, which is considered by many mathematicians to be the most important
unsolved problem in
pure mathematics. The Riemann zeta function plays a pivotal role in
analytic number theory and has applications in
physics,
probability theory, and applied
statistics.

Portal:Mathematics/Selected picture/14

Credit:
Pbroks13

The four **
conic sections** arise when a
plane cuts through a
double cone in different ways. If the plane cuts through parallel to the side of the cone (case 1), a *
parabola* results (to be specific, the parabola is the shape of the planar
graph that is formed by the set of points of intersection of the plane and the cone). If the plane is perpendicular to the cone's
axis of symmetry (case 2, lower plane), a *
circle* results. If the plane cuts through at some angle between these two cases (case 2, upper plane) — that is, if the angle between the plane and the axis of symmetry is larger than that between the side of the cone and the axis, but smaller than a
right angle — an *
ellipse* results. If the plane is parallel to the axis of symmetry (case 3), or makes a smaller positive angle with the axis than the side of the cone does (not shown), a *
hyperbola* results. In all of these cases, if the plane passes through the point at which the two cones meet (the vertex), a *
degenerate conic* results. First studied by the
ancient Greeks in the 4th century
BCE, conic sections were still considered advanced mathematics by the time
Euclid (
fl.
c. 300 BCE) created his *
Elements*, and so do not appear in that famous work. Euclid did write a work on conics, but it was lost after
Apollonius of Perga (
d. c. 190 BCE) collected the same information and added many new results in his *
Conics*. Other important results on conics were discovered by the medieval Persian mathematician
Omar Khayyám (d. 1131
CE), who used conic sections to solve
algebraic equations.

Portal:Mathematics/Selected picture/15

Credit:
Elphaba

This is a modern reproduction of the first published image of the **
Mandelbrot set**, which appeared in 1978 in a technical paper on
Kleinian groups by
Robert W. Brooks and Peter Matelski. The Mandelbrot set consists of the points c in the
complex plane that generate a bounded sequence of values when the recursive relation *z*_{n+1} = *z*_{n}^{2} + *c* is repeatedly applied starting with *z*_{0} = 0. The boundary of the set is a highly complicated
fractal, revealing ever finer detail at increasing magnifications. The boundary also incorporates
smaller near-copies of the overall shape, a phenomenon known as quasi-
self-similarity. The
ASCII-art depiction seen in this image only hints at the complexity of the boundary of the set. Advances in computing power and computer graphics in the 1980s resulted in the publication of
high-resolution color images of the set (in which the colors of points outside the set reflect how quickly the corresponding sequences of complex numbers diverge), and made the Mandelbrot set widely known by the general public. Named by mathematicians
Adrien Douady and
John H. Hubbard in honor of
Benoit Mandelbrot, one of the first mathematicians to study the set in detail, the Mandelbrot set is closely related to the
Julia set, which was studied by
Gaston Julia beginning in the 1910s.

Portal:Mathematics/Selected picture/16

Portal:Mathematics/Selected picture/17

Credit:
Jkasd

This is a chart of all
prime knots having seven or fewer
crossings (not including mirror images) along with the
unknot (or "
trivial knot"), a closed loop that is not a prime knot. The knots are labeled with
Alexander-Briggs notation. Many of these knots have special names, including the
trefoil knot (3_{1}) and
figure-eight knot (4_{1}). **
Knot theory** is the study of
knots viewed as different possible
embeddings of a
1-sphere (a
circle) in three-dimensional
Euclidean space (**R**^{3}). These mathematical objects are inspired by
real-world knots, such as knotted ropes or
shoelaces, but
don't have any free ends and so cannot be untied. (Two other closely related mathematical objects are
braids, which can have loose ends, and
links, in which two or more knots may be intertwined.) One way of distinguishing one knot from another is by the number of times its two-dimensional depiction crosses itself, leading to the numbering shown in the diagram above. The prime knots play a role very similar to
prime numbers in
number theory; in particular, any given (non-trivial) knot can be uniquely expressed as a "
sum" of prime knots (a series of prime knots spliced together) or is itself prime. Early knot theory enjoyed a brief period of popularity among physicists in the late 19th century after
William Thomson suggested that atoms are knots in the
luminiferous aether. This led to the first serious attempts to catalog all possible knots (which, along with links, now number in the billions). In the early 20th century, knot theory was recognized as a subdiscipline within
geometric topology. Scientific interest was resurrected in the latter half of the 20th century by the need to understand knotting problems in
organic chemistry, including the behavior of
DNA, and the recognition of connections between knot theory and
quantum field theory.

Portal:Mathematics/Selected picture/18

Credit:
Wridgers

A **
Klein bottle** is an example of a closed
surface (a two-dimensional
manifold) that is
non-orientable (no distinction between the "inside" and "outside"). This image is a representation of the object in everyday three-dimensional space, but a true Klein bottle is an object in
four-dimensional space. When it is
constructed in three-dimensions, the "inner neck" of the bottle curves outward and intersects the side; in four dimensions, there is no such self-intersection (the effect is similar to a
two-dimensional representation of a cube, in which the edges seem to intersect each other between the corners, whereas no such intersection occurs in a true
three-dimensional cube). Also, while any real, physical object would have a thickness to it, the surface of a true Klein bottle has no thickness. Thus in three dimensions there is an inside and outside in a colloquial sense: liquid forced through the opening on the right side of the object would collect at the bottom and be contained on the inside of the object. However, on the four-dimensional object there is no inside and outside in the way that a
sphere has an inside and outside: an unbroken curve can be drawn from a point on the "outer" surface (say, the object's lowest point) to the right, past the "lip" to the "inside" of the narrow "neck", around to the "inner" surface of the "body" of the bottle, then around on the "outer" surface of the narrow "neck", up past the "seam" separating the inside and outside (which, as mentioned before, does not exist on the true 4-D object), then around on the "outer" surface of the body back to the starting point (see the light gray curve on
this simplified diagram). In this regard, the Klein bottle is a higher-dimensional analog of the
Möbius strip, a two-dimensional manifold that is non-orientable in ordinary 3-dimensional space. In fact, a Klein bottle
can be constructed (conceptually) by "gluing" the edges of two Möbius strips together.

Portal:Mathematics/Selected picture/19

Credit:
Cburnett

This logic diagram of a
full adder shows how **
logic gates** can be used in a
digital circuit to add two
binary inputs (i.e., two input
bits), along with a carry-input bit (typically the result of a previous addition), resulting in a final "sum" bit and a carry-output bit. This particular circuit is implemented with two
XOR gates, two
AND gates and one
OR gate, although equivalent circuits may be composed of only
NAND gates or certain combinations of other gates. To illustrate its operation, consider the inputs A = 1 and B = 1 with C_{in} = 0; this means we are adding 1 and 1, and so should get the number 2. The output of the first XOR gate (upper-left) is 0, since the two inputs do not differ (1 XOR 1 = 0). The second XOR gate acts on this result and the carry-input bit, 0, resulting in S = 0 (0 XOR 0 = 0). Meanwhile, the first AND gate (in the middle) acts on the output of the first gate, 0, and the carry-input bit, 0, resulting in 0 (0 AND 0 = 0); and the second AND gate (immediately below the other one) acts on the two original input bits, 1 and 1, resulting in 1 (1 AND 1 = 1). Finally, the OR gate at the lower-right corner acts on the outputs of the two AND gates and results in the carry-output bit C_{out} = 1 (0 OR 1 = 1). This means the final answer is "0-carry-1", or "10", which is the
binary representation of the number 2. Multiple-bit adders (i.e., circuits that can add inputs of 4-bit length, 8-bit length, or any other desired length) can be implemented by
chaining together simpler 1-bit adders such as this one. Adders are examples of the kinds of simple digital circuits that are combined in sophisticated ways inside computer
CPUs to perform all of the functions necessary to operate a digital
computer. The fact that simple electronic switches could implement
logical operations—and thus simple
arithmetic, as shown here—was realized by
Charles Sanders Peirce in 1886, building on the mathematical work of
Gottfried Wilhelm Leibniz and
George Boole, after whom
Boolean algebra was named. The first modern electronic logic gates were produced in the 1920s, leading ultimately to the first digital, general-purpose (i.e.,
programmable) computers in the 1940s.

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Portal:Mathematics/Selected picture/21

Credit:
M.qrius

The **
sieve of Eratosthenes** is a simple
algorithm for finding all
prime numbers up to a specified maximum value. It works by identifying the prime numbers in increasing order while removing from consideration
composite numbers that are multiples of each prime. This animation shows the process of finding all primes no greater than 120. The algorithm begins by identifying 2 as
the first prime number and then crossing out every multiple of 2 up to 120. The next available number, 3, is the next prime number, so then every multiple of 3 is crossed out. (In this version of the algorithm, 6 is not crossed out again since it was just identified as a multiple of 2. The same optimization is used for all subsequent steps of the process: given a prime p, only multiples no less than *p*^{2} are considered for crossing out, since any lower multiples must already have been identified as multiples of smaller primes. Larger multiples that just happen to already be crossed out—like 12 when considering multiples of 3—*are* crossed out again, because checking for such duplicates would impose an unnecessary speed penalty on any real-world implementation of the algorithm.) The next remaining number, 5, is the next prime, so its multiples get crossed out (starting with 25); and so on. The process continues until no more composite numbers could possibly be left in the list (i.e., when the square of the next prime exceeds the specified maximum). The remaining numbers (here starting with 11) are all prime. Note that this procedure is easily extended to find primes in any given
arithmetic progression. One of several
prime number sieves, this ancient algorithm was attributed to the
Greek mathematician
Eratosthenes (
d.
c. 194
BCE) by
Nicomachus in his first-century (
CE) work *
Introduction to Arithmetic*. Other more modern sieves include the
sieve of Sundaram (1934) and the
sieve of Atkin (2003). The main benefit of sieve methods is the avoidance of costly
primality tests (or, conversely,
divisibility tests). Their main drawback is their restriction to specific ranges of numbers, which makes this type of method inappropriate for applications requiring very large prime numbers, such as
public-key cryptography.

Portal:Mathematics/Selected picture/22

Credit:
Schutz

Portal:Mathematics/Selected picture/23

Credit:
Fangz (original uploader)

This is
Francis Galton's original 1889 drawing of three versions of a "
bean machine", now commonly called a "Galton box" (another name is a *
quincunx*), a real-world device that can be used to illustrate the **
de Moivre–Laplace theorem** of
probability theory, which states that the
normal distribution is a good approximation to the
binomial distribution provided that the number of repeated "trials" associated with the latter distribution is sufficiently large. As the "bean" (i.e., a small ball) falls through the box (the design of which is quite similar to the popular Japanese game
Pachinko), it can fall to the left or right of each pin it approaches. Since each lower pin is centered horizontally beneath a pair of higher pins (or a higher pin and the side of the box), the bean has the same probability of falling either way, and each such outcome is approximately independent of the others. Each row of pins thus corresponds to a
Bernoulli trial with "success" probablility 0.5 ("success" is defined as falling a particular direction—say, to the right—each time). This makes the final position of the bean at the bottom of the box the sum of several approximately-independent
Bernoulli random variables, and therefore approximately a random observation from a binomial distribution. (Note that because the bean may reach the side of the box and at that point only be able to fall in one direction, this sequence of Bernoulli random variables might be interrupted by a *non-random* movement back towards the center; this would not be a problem if the box were wide enough to prevent the bean from reaching the side of the box, as in the top half of Fig. 8—see also
this photograph.) The box on the left, in Fig. 7, has 23 rows of pins (not counting the first row which is positioned in such a way that the bean always passes between two particular pins) and a final row of slots, so the number of trials in that case is 24. This is large enough that the resulting columns of beans collected at the bottom of the box show the classic "bell curve" shape of the normal distribution. While a level box gives a probability of 0.5 to fall either way at each pin, a tilted box results in asymmetric probabilities, and thus a
skewed distribution (see
this other photograph). Published in 1738 by
Abraham de Moivre in the second edition of his textbook *
The Doctrine of Chances*, the de Moivre–Laplace theorem is today recognized as a special case of the more familiar
central limit theorem. Together these results underlie a great many
statistical procedures widely used today in science, technology, business, and government to analyze data and make decisions.

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Portal:Mathematics/Selected picture/25

Credit:
Cmglee

Nicomachus's theorem states that the sum of the cubes of the first n
natural numbers is the square of the sum of the first n natural numbers. This result is generalized by
Faulhaber's formula, which gives the sum of pth powers of the first n natural numbers. The special case of Nicomachus's theorem can be proved by
mathematical induction, but a more direct proof can be given which is illustrated by a
proof without words, pictured here.

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Portal:Mathematics/Selected picture/27

Credit:
User:LucasVB

An animation showing how an obliquely cut
torus reveals a pair of intersecting circles known as **
Villarceau circles**, named after the French astronomer and mathematician
Yvon Villarceau. These are two of the four circles that can be drawn through any given point on the torus. (The other two are oriented horizontally and vertically, and are the analogs of lines of
latitude and
longitude drawn through the given point.) The circles have no known practical application and seem to be merely a curious characteristic of the torus. However, Villarceau circles appear as the fibers in the
Hopf fibration of the
3-sphere over the ordinary
2-sphere, and the Hopf fibration itself has interesting connections to
fluid dynamics,
particle physics, and
quantum theory.

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In
projective geometry, **
Desargues' theorem** states that two triangles are in
perspective axially
if and only if they are in perspective centrally. Lines through the triangle sides meet in pairs at
collinear points along the axis of perspectivity. Lines through corresponding pairs of
vertices on the triangles meet at a point called the center of perspectivity.

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An animated geometric
proof of the **
Pythagorean theorem**, which states that among the three sides of a
right triangle, the square of the
hypotenuse is equal to the sum of the squares of the other two sides, written as *a*^{2} + *b*^{2} = *c*^{2}. A large square is formed with area *c*^{2}, from four identical right triangles with sides *a*, *b* and *c*, fitted around a small central square (of side length *b* − *a*). Then two rectangles are formed with sides *a* and *b* by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas *a*^{2} and *b*^{2}, which together must have the same area as the initial large square. This is a somewhat subtle example of a
proof without words.

Portal:Mathematics/Selected picture/30

Credit:
User:JasonHise

A
three-dimensional projection of a **
tesseract** performing a
simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom. Also called an *8-cell* or *octachoron*, a tesseract is the
four-dimensional analog of the
cube (i.e., a 4-D
hypercube, or *4-cube*), where motion along the fourth dimension is often a representation for bounded transformations of the cube through
time. The tesseract is to the cube as the cube is to the
square. Tesseracts and other
polytopes can be used as the basis for the
network topology when linking multiple processors in
parallel computing.