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Pascal’s Theorem

Interesting creation from Gábor Damásdi, a Hungarian math student, currently doing his master degree at Eötvös Loránd University in Budapest. In his free time he likes to draw mathematical stuff like fractals, tillings, tessellations, polyhedrons and so on.
► Learn more about the author>>
http://szimmetria-airtemmizs.tumblr.com/about

This animation refers to Pascal's theorem, also known as the hexagrammum mysticum theorem.

In projective geometry, this theorem states that if six arbitrary points are chosen on a conic (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon (the red line in the animation below).

This theorem is a generalization of Pappus's (hexagon) theorem that is the special case of a degenerate conic of two lines.

Pascal's theorem- the dual of Brianchon's theorem- was formulated by Blaise Pascal in a note written in 1639 when he was 16 years old and published the following year as a broadside (printing) titled "Essay povr les coniqves. Par B. P."

► Animation source>>
http://szimmetria-airtemmizs.tumblr.com/post/161655210843/pascals-theorem-no-matter-how-you-choose-the-red

Further reading and references

► Projective Geometry>> https://en.wikipedia.org/wiki/Projective_geometry

► Pascal’s Theorem>> https://en.wikipedia.org/wiki/Pascal%27s_theorem

► Pappus's hexagon theorem>> https://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem

► Degenerate conic>> https://en.wikipedia.org/wiki/Degenerate_conic

► Broadside (printing)>> https://en.wikipedia.org/wiki/Broadside_(printing)

► Pascal's Theorem from Wolfram MathWorld>> http://mathworld.wolfram.com/PascalsTheorem.html

► Brianchon's Theorem>> http://mathworld.wolfram.com/BrianchonsTheorem.html

► Duality (projective geometry)>> https://en.wikipedia.org/wiki/Duality_(projective_geometry)


#mathematics, #art, #PascalTheorem, #processing, #symmetry, #geometry, #circle
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Turn, Magic Wheel

Another animation from Clayton Shonkwiler.

Something simple: three squares rotating around an axis. But the slightly unusual perspective makes it surely interesting.

Here's the code>> http://community.wolfram.com/groups/-/m/t/1030934

Clayton Shonkwiler is a mathematician and artist. He is interested in geometric models of physical systems; currently he is mostly focused on geometric approaches to studying random walks with topological constraints, which are used to model polymers.

► Go to:

- his website>> https://shonkwiler.org/

#Mathematics, #Geometry, #Animation
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Triakis

A stunning creation from Clayton Shonkwiler, mathematician (Colorado State University) and artist.

► Go to his website>> https://shonkwiler.org/

He wrote:

"Another in my collection of animations given by taking parallel cross sections of some shape, in this case the triakis icosahedron."

In geometry, the triakis icosahedron (or kisicosahedron) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.

► Go to the source, where you can get the code>>
http://community.wolfram.com/groups/-/m/t/795666

Further reading

► Cross section>> https://en.wikipedia.org/wiki/Cross_section_(geometry)

► Triakis icosahedron>> https://en.wikipedia.org/wiki/Triakis_icosahedron

► Archimedean solid>> https://en.wikipedia.org/wiki/Archimedean_solid

► Catalan solid>> https://en.wikipedia.org/wiki/Catalan_solid

► Truncated dodecahedron>> https://en.wikipedia.org/wiki/Truncated_dodecahedron


#Mathematics, #Animation, #Geometry, #CrossSections
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A Geometric Construction of the Parabola

► A Matthen's creation>>
http://blog.matthen.com/post/6000287357/a-geometric-construction-of-the-parabola-the-blue

The blue point is called the focus, and the horizontal line is the directrix. The blue lines show all the points which are at an equal distance from the red point and the blue focus point.
The point of the blue line directly above the red dot contributes to the parabolic curve, because the parabola is defined as the set of all points equidistant to the focus and the directrix.

► Go to the code>> https://pastebin.com/k5FrFVfp

Further reading

► Constructing a Parabola>> http://jwilson.coe.uga.edu/EMAT6680Fa07/O%27Kelley/Assignment%206/Parabola.html

► Methods of constructing parabolas>> http://www3.ul.ie/~rynnet/swconics/SP.htm

#Mathematics, #GeometricCurves, #Geometry, #ConicSections, #Animations
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Anything Goes

A creation from Clayton Shonkwiler, mathematician and artist.

He wrote:

"It's an obvious but kind of cool fact that every quadrilateral tiles the plane. Building on some ideas from Cut the Knot, I decided to write a function which would, given the edges of a quadrilateral (as vectors), produce a tessellation. Here's what I ended up with."

► Go to the source, where you can get the code>> http://community.wolfram.com/groups/-/m/t/865171

Further reading

► Simple Quadrilaterals Tessellate the Plane>>
http://www.cut-the-knot.org/Curriculum/Geometry/QuadTessellation.shtml


#Mathematics, #Art, #Tesselation, #QuadrilateralTiles, #Geometry
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Hamiltonian (Circuit of Octahedra)

An interesting creation from Clayton Shonkwiler, mathematician (Colorado State University) and artist.
► Go to his website>> https://shonkwiler.org/

This animation shows 6 octahedra centered at the 6 vertices of a large octahedron which trace out a Hamiltonian cycle on the skeleton of the big octrahedron.

► Go to this link for more information and to get the code>>
http://community.wolfram.com/groups/-/m/t/976338

Further reading

► Hamiltonian Cycle>> http://mathworld.wolfram.com/HamiltonianCycle.html

► Octahedron>> http://mathworld.wolfram.com/Octahedron.html

#Mathematics, #Geometry, #HamiltonianCycle, #Octahedron
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Penrose Triangle (or ImpossibleTribar)

The Penrose triangle was first created by the Swedish artist Oscar Reutersvärd in 1934. The psychiatrist Lionel Penrose and his mathematician son Roger Penrose independently devised and popularized it in the 1950s, describing it as "impossibility in its purest form".

It is an impossible object (also known as an impossible figure or an undecidable figure) that is a type of optical illusion. It consists of a two-dimensional figure which is instantly and subconsciously interpreted by the visual system as representing a projection of a three-dimensional object. 2D figure is subconscionsly interpreted as 3D object although such object can not exist.

It is featured prominently in the works of artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it.

Impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline.

► Animation source>> http://i.imgur.com/JYDFobl.gifv

Further reading and references

► Impossible object>> https://en.wikipedia.org/wiki/Impossible_object

► Penrose triangle>> https://en.wikipedia.org/wiki/Penrose_triangle

► Optical illusion>> https://en.wikipedia.org/wiki/Optical_illusion


#Mathematics, #Psychology, #Art, #Animation, #Tribar
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Something dating back to three years ago...
Kaleidoscope

Animated gif made by me using GeoGebra software.

It's useful for students to study  simmetries.

#kaleidoscope #GeoGebra #mathematics
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Play of Circles

This animation from Matt Henderson (aka Matthen) dates back to 2013. I found it by serendipity, and I'm sharing it because there is an interesting, simple problem to solve.

The author writes:

Two touching identical circles have the same area as the negative space they create in a circumscribing larger circle. That allows us to create this gif, where the circles transform without changing area. [can you prove the first sentence?]

► Source>> http://blog.matthen.com/post/55197665403/two-touching-identical-circles-have-the-same-area

#Mathematics, #Gifs, #Geometry, #Circles
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