A self avoiding walk is a path through a graph where no vertex is visited more than once. One problem is to consider the cartesian lattice, and the paths from \((0,0)\) to \((m,n)\). If we only allow paths in the “positive direction”, so that from \((i,j)\) the path can only proceed to \((i+1,j)\) or \((i,j+1)\) we …

Read More

The Asian Technology Conference in Mathematics is a friendly, genial conference which has been held almost every year since 1996. All the proceedings are freely available on that website, but there is no search method. If you want to find out if anybody has presented a paper about, say, teaching linear algebra with …

Read More

Recall from the previous post that the Weierstrass-Durand-Kerner (WDK) method finds all roots of a polynomial equation in a manner similar to Newton’s method, and so converges quadratically. The Aberth-Ehrlich method (more commonly called Aberth’s method), is similar, but converges cubically, and as we will see, is …

Read More

This is a method for finding all the roots of a polynomial simultaneously, by applying a sort of Newton-Raphson method. It gets its name from everybody associated wth it: Weierstrass published a version of the algorithm in 1891, and then rediscovered (independently) by Durand in 1960 and Kerner in 1966, as you can see …

Read More

Recap, and background Two posts ago we showed how, given four points in the plane in general position, but with a few restrictions, it was possible to find two parabolas through those points. We used computer algebra. The steps were: Create four equations

\[ (Ax_i+By_i)^2+Cx_i+Dy_i+E=0 \]

for each of the four …

Read More

It is (well?) known that if \(x = at^2+bt+c\) and \(y=pt^2+qt+r\), then

\[ (Ax+By)^2+Cx+Dy+E=0 \]

where

\[\begin{aligned} A&=p\\ B&=-a\\ C&=qv_2-2pv_1\\ D&=-bv_2+2av_1\\ E&=v_1^2-v_2v_3 \end{aligned}\]

with \(\langle v1, v2, v3\rangle =\langle a,b,c\rangle \times \langle p,q,r\rangle\); that is, the \(v_i\) values are …

Read More

Introduction It is (or should be) well known that a parabola has the cartesian form

\[ (Ax+By)^2+Cx+Dy+E = 0. \]

This looks as though there are five values needed, but we can divide through in such a way as to make any of the coefficients we like equal to 1:

\[ (Px+Qy)^2+Rx+Sy+1 = 0. \]

and so we see that only four …

Read More

Although the method is simple to describe, the algebra becomes messy when written in full generality. For example, suppose we use the second method, with three points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) none of which are at the origin. The three equations are \begin{gather} (Ax_1+By_1)^2+Cx_1+Dy_1=0\ …

Read More

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 …

Read More

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 \(p_0\) to …

Read More