# Mathematical Art Gallery

This page is a sub-page of the page on our Mathematical Explainatorium.

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Related KMR-pages:

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Related sources of information:

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Escher’s Angels and Devils moving in the Riemann-Poincaré disk:

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“Heaven and Hell” in hyperbolic space:

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The Feminine Archetypical Form(ula): $\, z \, = \, \nabla \cdot \dfrac{\mathbf{V}}{| \, \mathbf{V} \, |} \; , \;$ where $\; \mathbf{V} = \begin{bmatrix} y \\ x \\ z \end{bmatrix}$.

Vector Analysis
Vektoranalys (in Swedish)

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Devil transformed by the complex exponential function: $\, w \, = \, e^{z + \textcolor{red}{p}} \,$:

Devil transformed by complex sin: $\, w \, = \, \sin(z + \textcolor{red}{p}) \,$:

Conformal Mappings (Ambjörn Naeve on YouTube)
Conformal Face Mapping
Conformal Mapping
Geometric Numbers
Complex Numbers

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Complex Multiplication:

Complex Spiralization:
(this video was created by mistake, while trying to simulate complex multiplication)

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Trigonometry:

The Sin and Cos functions:

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A Trigonometric Inequality:

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Stereographic projection of a spherical circle:
[Stereographic projection means projection from the north pole of the sphere
onto the tangent plane at its south pole]:

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Möbius Transformations Revealed

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Reflecting a plane wavefront in a saddle surface turns it into a Dupin Cyclide:

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Parallel wave fronts of a hyperbolic paraboloid:

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Double-Cylindrical Point Focusing Mirrors

A practical realization of the DC-Point Focus:
(by Thomas Elofsson, 1989):

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Confocal Quadrics 2 – – Elliptic Coordinates – Intersect XY-plane:

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The Tangent Developable of a Circular Helix:

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Inversion of a Tangent Surface (with its Principal Net):

Inversion of a Tangent Surface (with its Osculating Planes):

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The Rectifying Developable of a Circular Helix:

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Isometric Deformation: from Catenoid to Helicoid and back again:

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Parametric Surface – approximate Weingarten Map :

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Circular helix – square profile Monge surface:

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PSTC surface:

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CircularHelix-Ellipse-PSTC.guts (n = 2.2):

CircularHelix-Ellipse-PSTC.guts (movie):

CircularHelix-Ellipse-PSTC.guts (movie 2):

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Trig fractals (Book 2):

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Tesseract = 4-Dimensional Cube:

Triact = 3-Dimensional Cube:

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The complex exponential function

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A planar electromagnetic wave:

The electric part of the wave: $\, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \,(\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, \omega \, t)} \,$

The magnetic part of the wave: $\, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \, (\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, (\omega \, + \, \pi/2) \, t)} \,$

The entire wave: $\, E_m(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, + \, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \,$

Its Poynting vector : $\, S \, = \, \frac{1}{{\mu}_0} \, E \, \times \, B$

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Conceptual background:

Historical background:

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Projective pencil-to-pencil transformation:

Dual configuration:

Projective range-to-range transformation:

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Affine approximation of a function from R^2 to R:

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Plane Curves:

The Evolute of an Ellipse:

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Cycloids and Trochoids

Cycloid and Trochoid – orbits:

The interactive simulation that generated this movie.
Drag the blue point at the bottom horizontally to change the Trochoid.

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CycloidEvolute – Involutes:

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Inversion of a polar grid:

Interactive simulation of 1/z of a polar grid.
This is very similar to inversion, which would be 1/Conjugate(z).

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Devil-inversion-xy.anim (or, rather, Devil transformed by 1/z):

The interactive simulation that created this movie.
Drag the devil to study the dynamics of the transformation. Note the difference from the video clip. The interactive simulation is a true inversion.

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EpiTrochoid-center-inverse:

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Ellipsefocus-pedal:

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Circle-internal-point-negative-pedal:

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Cissoid – from a circle, a free point, and a free line:

The distance between the green point and the blue point (on the black line) is equal to
the distance between the red point and the black point.

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A generic Conchoid:

The interactive simulation that generated this movie.
The green point and the black point can be moved.

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Strophoid – from a circle and two generic points:

The interactive simulation that created this movie.
The green points can be moved.

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Ellipse – Focal Ray Reflect:

Ellipse – Focal Wave Reflect:

Ellipse – Focal Wave/Ray Reflect:

/////// Caustic Curves:

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Ellipse – near-focus caustic:

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Ellipse-Vertex-Caustic:

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Semi-circle parallel light rays real caustic:

Semi-circle parallel light rays virtual caustic:

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Taylor Expansion

The Taylor polynomial of degree 1 for the function f(x, y) at the point (a, b):
(also known as The Affine Approximation):

${T_{aylor}^1 (f)}_{(a,b)} = f(a,b) + \frac{\partial f}{\partial x}_{(a,b)} (x-a) + \frac{\partial f}{\partial y}_{(a,b)} (y-b)$

Interactive simulation of Taylor expansion of order 1.
Drag the red point in the left window to move the point of expansion.

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The Taylor polynomial of degree 2 for the function f(x, y) at the point (a, b):

${T_{aylor}^2(f)}_{(a,b)} = {T_{aylor}^1 (f)}_{(a,b)} + \dfrac{1}{2} (\frac{\partial^2 f}{\partial x^2}_{(a,b)} (x-a)^2 + 2 \frac{\partial^2 f}{\partial x \partial y}_{(a,b)} (x-a)(y-b) + \frac{\partial^2 f}{\partial y^2}_{(a,b)} (y-b)^2)$

Interactive simulation of Taylor expansion of order 2.
Drag the red point in the left window to move the point of expansion.

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The Taylor polynomial of degree 3 for the function f(x, y) at the point (a, b):

${T_{aylor}^3(f)}_{(a,b)} = {T_{aylor}^2 (f)}_{(a,b)} + \dfrac{1}{6} (\frac{\partial^3 f}{\partial x^3}_{(a,b)} (x-a)^3 + 3 \frac{\partial^3 f}{\partial x^2 \partial y}_{(a,b)} (x-a)^2 (y-b) + 3 \frac{\partial^3 f}{\partial x \partial y^2}_{(a,b)} (x-a)(y-b)^2 + \frac{\partial^3 f}{\partial y^3}_{(a,b)} (y-b)^3)$

Interactive simulation of Taylor expansion of order 3.
Drag the red point in the left window to move the point of expansion.

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The Taylor polynomial of degree 4 for the function f(x, y) at the point (a, b):

Interactive simulation of Taylor expansion of order 4.
Drag the red point in the left window to move the point of expansion.

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The Taylor polynomials of degrees 1 and 2 for the function f(x, y) at the point (a, b):

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The Taylor polynomials of degrees 1 and 3 for the function f(x, y) at the point (a, b):

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The Taylor polynomials of degrees 1 and 4 for the function f(x, y) at the point (a, b):

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The Taylor polynomials of degrees 1, 2, 3 for the function f(x, y) at the point (a, b):

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The Taylor polynomials of degrees 1, 2, 3, 4 for the function f(x, y) at the point (a, b):

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