the Lie Group E8

L Spiral.

This is a generalization of a diagram I saw from d3lt4 on Tumblr: It is constructed by rotating and dilating the L shape to fit in with itself, rotated 90 degrees. There are some interesting properties that develop based on the height/width ratio and the thickness of the L-shape. At first I thought D3lt4’s diagram was based on an L from a square with thickness half of the width, but instead it is Golden.

The last image is close to real Golden ratio proportions.

Envelope and String Art

In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two “adjacent” curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

Image:

Gifs: Shared at Mathani.tumblr - Lightprocesses.tumblr& Envelope on Wikipedia.

Hand embroidered cards by Les Fils Rouges.

How do you make: String art by Lay Ágota.

Envelop doodle by Dan MacKinnon.

Mathematics in 3d

Herwig Hauser Classic(P2) - Authors | Herwig Hauser

Nepali: Let us look at the defining equation (xy−z3−1)2 = (1−x2−y2)3 of Nepali. The symmetry between x and y is enforced by the quadratic polynomial x2+y2, which is rotation symmetrical in contrast to the monomial xy. Sections with horizontal planes z = c yield closed curves being almost circles. The simultaneous occurrence of squares and third powers produces the tapering at the top.The lateral boundary curve of Nepali is not an exact circle but is arching up and down like the brim of a hat. Its projection to the horizontal xy plane, however, is a circle as can be seen from the top view. The surface shown is bounded; hence there was no need to limit the view by sphere intersection. This fact can be directly derived from the formula by accurate analysis.

Zeck (Tick): The simple Tick equation x2+y2 = z3(1−z) fully dictates the geometry just as with the other surfaces; that means, both the singular points and the outer shape, the curvature and the extension are clearly defined by the four monomials x2, y2, −z3 und z4. As a result, the formula is a very efficient way to codify forms which appear complicated. However, the geometric information cannot always be read from the formula. The local shape of the surface near a given point can be explicitly defined, in most cases; the techniques of local analytic geometry have a good effect. Defining the global structure requires much more efforts and cannot always be satisfactorily accomplished.

Kreisel(Spinning Top): To illustrate the shape of a spinning top as a solution set of a simple algebraic equation needs some inventive mathematics! It is, indeed, not clear mathematically, at first sight, what this equation must look like because different equations can provide similar looking forms. The image shows the equation 60(x2+y2)z4 = (60−x2−y2−z2)3. The rotation symmetry round the vertical z axis is identified by x and y occurring in the quadratic polynomial x2+y2. The third power on the right hand side is necessary to produce the two points. And as z occurs only in even powers, the surface is symmetrical with respect to the reflection in the horizontal xy plane. The parameter 60 in the equation is used for aesthetic reasons.

 Eistüte (Cone):  The horizontal section through Cone is a so-called rosette curve: A small wheel rolls round the interior of an annular body while a pencil fixed to the wheel is drawing a curve (like a Spirograph).  Different curves are formed depending on the ratio of the two radii. The curve closes if the ratio is a rational number.  In our case it is the four- leaf clover. Our cone has the advantage that four scoops of ice-cream will fit in!

Helix: The Lemniscate is the plane curve with the equation y4+z2 = y2. It results from the circle y2+z2 =1 by substituting z by z/y. Geometrically, the substitution corresponds to a distortion of the circle to a figure of eight loop. If y4+z2 = y2 is conceived as equation with three variables x, y and z (where x is hidden) then the solution set is a surface in the three-dimensional space, i. e. the cylinder above the lemniscate. The Helix equation y4+x2z2 = y2 is yielded by substituting z by xz. From the geometrical view, this construction is a kind of folding. The symmetry with respect to x and z is clearly visible. For the final formula we added the factors 2 and 6 to slightly stretch Helix. The singular locus is a crossing of straight lines. The sections of Helix with the planes x = c or z = c are lemniscates for c ≠ 0, whereas the sections y = c are hyperbola pairs.

Himmel & Hölle (Heaven and Hell): A piece of paper is folded and is held from beneath such that you can put your four fingers in the four corners so formed. By spreading your fingers the figure opens in two different ways so that two of the four inner sides can be seen at a time.

This figure reminds us of the popular children’s fortune teller game where you have to predict which colour will show up – blue for heaven and red for hell (hence the name!)

By adding up the squares at y and z you get the highest exponent 4. This is called an equation of the 4th degree. The higher the degree the more complicated it is to calculate the surface.

A  Bézier curve is a mathematically defined curve used in computer graphics and similar applications. The curve is defined by four points: the initial position and the terminating position (which are called “anchors”) and two separate middle points (which are called “handles”). The shape of a Bézier curve can be altered by moving the handles.

Unfolding of Spherical Surfaces. Make paper sphere models with approximate patterns. This patterns are really wonderful. Because, determining how to unfold a polyhedron or a sphere into a net is tricky. For example, determining how to unfold a polyhedron, cuts cannot be made along all edges that surround a face or the face will completely separate. Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. In fact, the edges that must be cut corresponds to a special kind of graph called a spanning tree of the skeleton of the polyhedron (Malkevitch). Source.

See more: Unfolding at Mathworld.wolfram.com &  Dymaxion Map on Wikipedia.

Each dot is only moving in a straight line, but is created by balls moving in circles through 3 dimensional space. 

Integral and the difference between: The Riemann-Darboux approach & The Lebesgue approach. Riemann-Darboux’s integration (in blue) and Lebesgue integration (in red).  To get some intuition about the different approaches to integration, let us imagine that it is desired to find a mountain’s volume (above sea level). The Riemann-Darboux approach: Divide the base of the mountain into a grid of 1 meter squares. Measure the altitude of the mountain at the center of each square. The volume on a single grid square is approximately 1 m2 × (that square’s altitude), so the total volume is the sum of the altitudes.

The Lebesgue approach: Draw a contour map of the mountain, where adjacent contours are 1 meter of altitude apart. The volume of earth contained in a single contour is approximately 1 m × (that contour’s area), so the total volume is the sum of these areas. Professor Gerald B.Folland  summarizes the difference between the Riemann and Lebesgue approaches thus: “to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals”, while in the Lebesgue integral, “one is in effect partitioning the range of f “ 

( Gerald B.Folland (1999). Real analysis: Modern techniques and their applications. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. xvi+386. )

See more at Integral on Wikipedia  - Riemann integral   - Lebesgue integration