Mathematics is used throughout the world as an essential tool in many fields. There are a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Presented below are some interesting animations in relation to Mathematics.

### 1) Set Theory

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It is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Set theory begins with a fundamental binary relation between an object *o* and a set *A*. If *o* is a member (or element) of *A*, write *o* ∈ *A*. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set *A* are also members of set *B*, then *A* is a subset of *B*, denoted *A* ⊆ *B*. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not. From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule this out, the term proper subset is defined. *A* is called a proper subset of *B* if and only if *A* is a subset of *B*, but *B* is not a subset of *A*.

### 2) Mapping a Curve on Cartesian plane into Polar Coordinates

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A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, is mapped onto . Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate.

### 3) A Hyperboloid

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The animation shows the movement of a hyperboloid through a hyperbola. The bar in the gif makes a hyperboloid and the curve the bar passes through is a hyperbola.

### 4) Pythagoras Theorem Proof

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### 5) A Convex Polygon with n-sides within a circle

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A convex polygon with n sides is drawn in a circle. Then divided into triangles. Finally circles are inscribed in the triangles. Obviously, triangulation is not unique; a 4-gon has two, a 5-gon five, a 6-gon 14 and so on, as continue the Catalan numbers. But the sum of the radii of the circles is constant, independent of how the n-gon is divided.

### 6) Evolute of an Ellipse

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In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. An ellipse (blue) and its evolute (green). The moving circle is the osculating circle to the ellipse, whose center is the center of curvature. It is also shown how the tangent line to the evolute is normal to the ellipse, i.e., the evolute is the envelope of the normals to the ellipse. The evolute of an ellipse is called an astroid.

### 7) Caustic

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Caustic is a phenomenon whereby a pattern is created by light rays when reflecting off a semi-circle. The rays are drawn imperfectly, with random variation as if they were drawn by hand. This type of pattern, the **caustic**, might be familiar from looking into a coffee cup in the Sun.

### 8) Fourier Series

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A Fourier series is a way to expand a periodic function in terms of sines and cosines. The Fourier series is named after Joseph Fourier, who introduced the series as he solved for a mathematical way to describe how heat transfers in a metal plate. The GIFs above show the 8-term Fourier series approximations of the square wave and the sawtooth wave.

### 9) Rubik’s Cube

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Ernö Rubik invented the Cube in the spring of 1974 in his home town of Budapest, Hungary. Algorithms exist for solving a cube from any specific position it is currently in, but typically there is no optimal solution; how quick a cube may be solved depends on the individual who is solving the cube and how quickly they are able to deal with the cube. The current world record for single time on a 3×3×3 Rubik’s Cube was set by Mats Valk of the Netherlands in March 2013 with a time of 5.55 seconds at the Zonhoven Open in Belgium. The cube consists of a grand number of combinations and permutations that it can be twisted into. The number of possible positions – permutations – of Rubik’s cube is

That is **43 quintillion** ways to possibly organize the cube.