These pages contain some Recreational Mathematics (Recmath) material with a focus on pattern and symmetry. The approach is a bit rambling and definitely short on analysis and proof. Hopefully the visual content makes up for some of the other shortcomings.
There are currently six sections about recreational mathematics topics like magic squares and polyominoes. There is a summary of each section below and the sections themselves can be accessed via the navigation bar buttons above or by following links in the summaries below.
The first section on Magic Square Patterns dates back to 1997 although the material itself was developed in the early 1970's. The most recent section on Turing Machine Patterns was put together late in 2007 after reading about the minimum Universal Turing Machine competition.
A new interactive version of the 'Projection Patterns' section has been added (July 2011).
Old URLs for these pages (from previous ISPs) were web.idirect.com/~recmath and ww3.sympatico.ca/diharper.
magic square patterns
This section explores patterns that can be generated when the numbers in order 4 pandiagonal magic squares are replaced with a set of geometric shapes. A magic square of order 4 is a square matrix of 16 numbers (usually the integers from 1 to 16) that have been arranged so that every row, column and diagonal adds up to the same sum. If the numbers from 1 to 16 are used then the sum for each row, column and diagonal should be 34. Pandiagonal means that the broken diagonals also add up to the magic sum of 34.
The patterns are generated by replacing each integer in the matrix with a unique geometric symbol.
In this section, the patterns are associated with polyominoes that have additional 'connection' information. The premise is that adjacent squares in a polyomino may or may not be connected as long as there is a 'connection path' between any two squares of the polyomino.
In the order 5 polyomino on the right, the two squares on the bottom row of the polyomino are not directly connected to each other although there is a connection path between them via the squares on the top row.
The Projection Patterns section just grew out of some doodles in which I was trying to find all of the projection drawings of six or seven cubes that had the same outline shape. These drawings are axonometric projections of type dimetric.
The figure on the right shows one of them. Click here to see another projection pattern within the same outline shape.
magic squares & polyominoes
Magic Squares & Polyominoes is about a link between polyominoes that can tile the plane and a construction method for Magic Squares called the De La Hire method by W. S. Andrews in his book Magic Squares and Cubes.
The link is demonstrated for order 5 magic squares but it will probably work for other orders greater than five.
Some pictures of integers as set constructions are in this section. It grew out of some attempts to make geometric drawings out of the way that natural numbers are constructed in axiomatic set theory.
turing machine patterns
This section has an implementation of some simple Turing Machines - machines with three tape symbols and two states and ones with two tape symbols and three states.
The motivation behind this was reading about the successful proof of the assertion that a three symbol/two state machine was universal and the interesting patterns used to illustrate the behaviour of the machines. These patterns result from stacking the sequence of one dimensional tape conditions into a two dimensional array and using different coloured squares for the tape symbols.
Last Updated on July 3rd. 2011
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