Recmath: Magic Squares, Polyominoes and more...
These pages contain some Recreational Mathematics (Recmath) material on topics like magic squares and polyominoes - an interest of mine since my school years back in the 1960s. Patterns and programming figure prominently since a lot of my school and work background was in IT, starting with Algol programs on paper tape. Some of the stuff is original or at least not seen by me before in books or online and the approach is a bit rambling and definitely short on analysis and proof.
There are currently seven sections and there is a summary of each section below together with a link to the page that contains the details of that topic.
The Web pages on Magic Square Patterns date back to 1997 although the material itself was developed in the early 1970's. The section on Turing Machine Patterns was put together late in 2007 after reading about the minimum Universal Turing Machine competition and a newer version of the 'Projection Patterns' section was added in July 2011.
The most recent section on a common pattern in the layout of the numbers in fourth order pandiagonal magic squares was added in 2015. At the same time the format of the Web pages for most of the sections was standardized
Old URLs for these pages (from previous ISPs) were web.idirect.com/~recmath and ww3.sympatico.ca/diharper.
fourth order pandiagonal square number layout
This section looks at a common structure that is present in the layout of the numbers of all fourth order pandiagonal squares: a cross formation consisting of the number 15 surrounded by the powers of 2 (i.e. 1, 2, 4 and 8) when the set of numbers 0 to 15 is used.
There are 24 ways (counting rotations and reflexions) of arranging 1, 2, 4 and 8 in a cross formation surrounding 15 and 16 options for placing the cross in a 4 by 4 square. 24*16 = 384 is also the exact number of fourth order pandiagonal squares and it can be shown that each cross arrangement corresponds to a unique pandiagonal magic square via a series of Sudoku-like logical steps.
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.