Roots of Unity

Sylvester's Stamp Puzzle

The stamp puzzle for denominations 4 c and 7 c is illustrate above. The lattice points on the plane z=4x+7y are shown in black. The integer(x,y) pairs are shown in red for given z. The yellow plane (z=17) has no integer solutions in this octant. It has integer solution (-1,3). Note 28 has solutions (7,0) and (0,4) and 56: (14,0),(0,8), (7,4)...

Equity

Complex Inversion

This post explores the relationship between the Riemann sphere representation of the extended complex plane and the complex plane. The 'north pole' is shown as a blue point. The points of the sphere are shown in red. Three points represent: z, 1/conjugate(z), 1/z. The 2D slider moves point on sphere. The black points are the stereographic projections of the points on the sphere onto the(complex) plane z=0 On the sphere the first mapping corresponds to reflection through the plane z=0. The second mapping is complex conjugation and corresponds to reflection in the plane y=0. The composite of these two reflections corresponds to an 180 degree rotation around the real axis(illustrated by the brown arc, noting the green line intersections the real axis).

Spontaneous and Fixed Singularities

This example comes from Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Bender and Orszag.

Triangles on Poincare Disk

The above CDF is uses locators to illustrate triangles in hyperbolic geometry as represented by the Poincare disk model.

The locators have not been constrained to the disk so errors arise once going outside. If someone knows how let me know. I have yet to find a way to colour the triangle.

This is a start of contemplation of this model.

CADILLAC Risk Score

Street View

This post is courtesy of Google Maps Javascript API. It is direct copy of the script but can be customised to suit ones needs.

TOD

Gambler's ruin

In this CDF I explore the Gambler's ruin. In the above simulation, 2 gamblers (A and B) each start with half the capital. At each game 1 unit of capital is bet by each. It is illustrated from the viewpoint of player A (player B is the complement).

100 'rounds' between player A and B with planned number of games given. Although the 'amounts' are unrealistic, it illustrates the properties. Green coloured plots are those runs of games that have not reached ruin for either gambler. Red plots have stopped. The discrete choices for probabilities and small capital values were chosen for computational (time) convenience.

The probability that gambler A is in neither of the absorbing states after the number of games was calculated using powers of the transition matrix (exploiting the sparse array structure (central band)) and initial vector based on equal entry capital of each gambler. This is compared with the frequency of green games in the simulation.

The long run probability and expectation are calculated using the formulas derived from solving the recursive relations with absorbing states as boundary conditions.

The observed frequencies count the number of red plots, the red plots in which A wins and the fraction of red plots that A wins.

Flavonoids

The heart.org have an interesting post regarding the flavonoids. This arises as the publication fraud issue with respect to Resveratol continues to reverberate in the research community.

The click on the planar molecular plot to be taken to the wikipedia entry. Hover over the image for the tooltip.

More fruit and vegetables, red wine, chocolate and nuts for all...while we wait for the evidence to come in...

Dropping a rock

This is a an illustrative demonstration of from the puzzle. I did this with little time. Options such as animating the rock, illustrating the effect of different densities of rock or object, e.g. floating objects and changing to assess and quantify the change are all options (a low priority on a CDF todo list).

Local maxima in permutations

This post accompanies How many bumps?.

The CDF is the graphical illustration of the permutations of numbers to 1 to n (in this case illustrating 2,3,...,6). The average is calculated using a function that counts the local maxima in each permutation and then averages across all permutations. This quickly becomes unwieldy. The simplicity of the solution and the efficiency and beauty of the derivation provided by Professor Blitzstein is now self-evident.

It would have been nice to color the local maxima: perhaps another post.

Network flow

This is a interactive illustration of a simple graph. The source is the left most node and the sink the right most node. The edge capacities are frame and blue The flows are frame with red text and yellow backgrounds. The subgraph highlighted in red illustrates the maximal flow.

The sliders change the capacities on the graph.

Hidden in plain sight

This may have been a relatively easy detection exercise but using Mathematica allowed visualization.

Peaucellier's linkage

This is an interactive CDF of the Peaucellier's linkage. A better illustration is on the Wolfram Demonstration project by Izidor Hafner.

I explore this linkage in Walking in a straight line.

How do you see the world

Someone introduced me to the joy of the xkcd site. I found this strip particularly entertaining. This prompted this post.

You can click on the map and you will be taken to the wikipedia entry about the projection (if there is one) or to a page that allows you to search. The Eckert projections do not have entries but really it is the fault of my code.

Newton's method: basins of attraction colored by root

In this post, I have modified the previous code to color points in the basins by the root they converge to and diverging points differently. The contrast (and consequent clarity) varies by color scheme.

Newton method: basins of attraction

This post is inspired by Invitation to Mathematics. The graphic is a ListDensityPlot of the number of iterations of Newton's method to find root till convergence (using length of the list from FixedPointList function) for points in the complex plane (the unit square). The polynomial is a cubic with roots zero, 1 and the parameter point. The roots are seen as the left and right lower corners and the point in the Slider2D. Gradient color schemes are used to represent the convergence. The basins of attraction are,therefore, illustrated as well as the set of points whose orbit diverges. In future posts, I aim to color the basins for the particular root.

Morse code