A bicentric heptagon is one for which all vertices lie on a circle, and for which all edges are tangential to another circle. If \(R\) and \(r\) are the radii of the outer and inner circles respectively, and \(d\) is the distance between their centres, there is an expression which relates the three values when a …

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Introduction

Poncelet's porism or Poncelet's closure theorem is one of the most remarkable results in plane geometry. It is most easily described in terms of circles: suppose we have two circles \(C\) and \(D\), with \(D\) lying entirely inside \(C\). Pick a point \(p_0\) on \(C\), and find the tangent from \(p_0\) …

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We have seen a number of different error diffusion matrices (and there are others we haven't discussed); the simplest of our matrices has been "Sierra Lite"

\[\frac{1}{4}\begin{bmatrix} 0& *& 2\\ 1& 1& 0 \end{bmatrix}\]

However, there is an even simpler one which seems to give very good …

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We have seen a number of different error diffusion matrices (and there are others we haven't discussed); the simplest of our matrices has been "Sierra Lite"

\[\frac{1}{4}\begin{bmatrix} 0& *& 2\\ 1& 1& 0 \end{bmatrix}\]

However, there is an even simpler one which seems to give very good …

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A totally different approach to dithering is error diffusion. Here, the image is scanned pixel by pixel. Each pixel is thresholded t0 1 or 0 depending on whether the pixel value is greater than 0.5 or not, and the error - the difference between the pixel value and its threshold - is diffuse across neighbouring …

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Image dithering, also known as half-toning, is a method for reducing the number of colours in an image, while at the same time trying to retain as much of its "look and feel" as possible. Originally this was required for newspaper printing, where no shades of grey were possible, and only black and white …

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In the previous post, we saw that a small change to the method of false position provided much faster convergence, while retaining its bracketing.

This was the Illinois method which is only one of a whole host of similar methods, some of which converge even faster.

And as a reminder, here's its definition, with a very …

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Such a long time since a last post! Well, that's academic life for you ...

If you look at pretty much any modern textbook on numerical methods, of which there are many, you'll find that the following methods will be given for the solution of a single non-linear equation \(f(x)=0\):

• direct iteration, also known as …

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Carroll originally invented his Doublets in 1877, they were published in "Vanity Fair" (the magazine, not the Thackeray novel) in 1879. Some years later, in an 1892 letter, Carroll added another rule: that permutations were allowed. This allows very neat chains such as:

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Apparently there's a version of Doublets (see previous post) which allows you to add or delete a letter each turn. Thus we can go from WHEAT to BREAD as

WHEAT, HEAT, HEAD, READ, BREAD

which is shorter than the ladder given in that previous post. However, we can easily adjust the material from that post to implement …

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