Mathematical Papers
A selection of mathematical articles and notes by Andrew Jobbings.
The PDF files on this page may be freely downloaded and printed for private or educational use.
The pleasure of learning and knowing, though not the keenest, is yet the least perishable of pleasures; the least subject to external things, and the play of chance, and the wear of time. And as a prudent man puts money by to serve as provision for the material wants of his old age, so too he needs to lay up against the end of his days provision for the intellect.
Published articles
Two semicircles fill half a circle
The Mathematical Gazette, Volume 95 (November 2011)
- Proves the surprising result that two touching semicircles, whose diameters are parallel chords of a circle, occupy half the area of the circle whatever the lengths of the chords are.
The Dance of the Foci Morphs into a Strophoid
The College Mathematics Journal, Volume 42 (September 2011)
- The intersection of a plane and a cone is a conic section and rotating the plane leads to a family of conics. What happens to the foci of these conics as the plane rotates? A classical result gives the locus of the foci as an oblique strophoid when the plane rotates about a tangent to the cone; rotation about a different axis gives a very different curve. This article relates the curves by analyzing the family obtained as the axis of rotation moves.
Proofs by dissection of a dodecagon
The Mathematical Gazette, Volume 95 (March 2011)
- Describes a solution to an Olympiad problem, which relates the problem to a result about the area of a dodecagon. Gives two proofs of this result by dissection.
Folding paper
The SMC Journal 40 (December 2010)
- The artistic side of paper-folding (origami) is well-known, but it is less well known that there are mathematical aspects too, and many of them. This article draws attention to some of this mathematics, and shows how folding paper may be helpful in teaching.
Sudoku is four-dimensional
The Mathematical Gazette, Volume 94 (July 2010)
- Shows that the standard 9×9 Sudoku puzzle is naturally a four-dimensional configuration.
- Enumerates the number of grids where six constraints are imposed rather than the usual three.
Geometry by numbers
The SMC Journal 39 (December 2009)
- Shows how to use complex numbers to prove geometrical results, including Varignon’s theorem, van Aubel’s theorem, Thébault’s first theorem and a remarkable result about the diagonals of a regular polygon.
A surprising relationship?
The Mathematical Gazette, Volume 91 (March 2007)
- Considers the accuracy of a scatterplot of the area versus perimeter of random rectangles.
Dissecting a triangle into a rectangle
The Mathematical Gazette, Volume 89 (November 2005)
- Describes a general procedure for dissecting any triangle into a rectangle with one side given.
Dissections
Maths Challenges News, Issue 9 (May 2001).
- Considers dissection problems and demonstrates how they may be solved by superimposing tessellations.
Playing with Dice
Maths Challenges News, Issue 8 (February 2001)
- Discusses unusual dice that behave like normal ones.
- Describes a set of non-transitive dice.
Parity
Maths Challenges News, Issue 5 (September 1999).
- Explains how a parity argument may be used to prove that something is impossible.
The volume of the n-ball
The Mathematical Gazette, Volume 82 (March 1998)
- Derives a formula for the volume of the n-dimensional ball.
Quadric quadrilaterals
The Mathematical Gazette, Volume 81 (July 1997)
- Demonstrates the necessary and sufficient conditions for the vertices of a quadrilateral to lie on the perimeter of a square.
Fair means
The Mathematical Gazette, Volume 81 (July 1997)
- Describes an approach to combining marks which in a well-defined sense is as fair as possible.
Chords, tangents and cubics
The Mathematical Gazette, Volume 79 (July 1995)
- Discusses the relationship between the roots of a cubic equation and the intersections of a cubic curve with a chord or a tangent.
A polyhedron and its volume
The Mathematical Gazette, Volume 68 (October 1984)
- Illustrates the polyhedron obtained when 24 symmetrically placed tetrahedra are removed from a cube.
- Calculates the volume of the polyhedron.
Miscellany
A collection of resources for teachers, unpublished articles and notes, made freely available.
How to fold simple shapes from A4 paper
- Starting with a single sheet of paper, how do you fold simple shapes like an equilateral triangle? Instructions are given for folding:
- a square
- an equilateral triangle
- a rhombus
- a regular hexagon
- a kite
- Explains why the methods work.
A graph with a unique Hamiltonian circuit
- A path which visits every vertex of a graph just once and returns to the starting point is known as a Hamiltonian circuit. How many Hamiltonian circuits does a given graph have?
- Shows that for a particular graph the Hamiltonian circuit is unique up to symmetry.
Chords and regions
- How many regions are created when chords are drawn in a circle?
- Shows the dangers of ‘pattern spotting’—deriving a formula from limited data—and gives two proofs of the correct result.
Trigonometry and the nonagon
- Stimulated by the article Three trigonometric results from a regular nonagon by David Miles and Chris
Pritchard in Mathematics in School (November 2008).
- Shows that each of the results discussed in the article is related to a more general result, though in very different ways.
The central region of the Mandelbrot set
- Shows that the central region of the Mandelbrot set is a cardioid.
A derivation of the quadratic formula
- A derivation of the quadratic formula from the relationships between the roots and coefficients (Vieta’s formulae).
BMO 2004/5 Round 1 question 5
- Describes a solution using continued fractions to a British Mathematical Olympiad problem.
Langley’s triangle problem
- A solution to Langley’s 20°/80°/80° isosceles triangle problem.
Newton’s derivation of Kepler’s second law
- Explains the elementary derivation that Newton gave of Kepler’s second law of planetary motion.
A generalisation of Pythagoras’ theorem
- A generalisation of Pythagoras’ theorem to squares of areas in three dimensions.
- The proof only involves the standard two-dimensional theorem.
Times of flight in a resisting medium
- An elementary proof that for vertical motion in a resisting medium the time of ascent is less than the time of descent, whatever the law of resistance.
© A K Jobbings 2004–2012
On this page
Clicking a link will scroll the page to the relevant section.
PDF documents
Free software to read and print PDF documents may be downloaded from Adobe: 
