Saturday, 10 September 2016

Magic torus coordinate and vector symmetries

Magic squares have fascinated mathematicians for centuries and they continue to do so today. However, many questions remain unanswered, and this study proposes a different perspective in order to shed new light on what magic squares are and how they work. Considering magic squares to be flattened partial viewpoints of convex or concave magic tori, the implied Gaussian surfaces require a modular arithmetic approach that is tested and analysed here.

Until today, most magic square construction methods use base squares, whilst "Magic torus coordinate and vector symmetries" proposes modular coordinate equations that not only define specific magic tori, but also generate their magic torus descendants.

The construction of Agrippa's traditional magic squares is analysed in detail for each of the seven planetary magic tori, and modular coordinate equations are defined that generate descendant tori throughout the respective higher-orders, whether they be odd, doubly-even, or singly-even.

The unique third-order magic torus and also a fifth-order pandiagonal torus of Latin construction are examined in detail. The modular coordinate equations that are defined generate an interesting variety of higher-odd-order descendant tori.

The study also explores the construction of the 12 different types of fourth-order magic tori, as well as the generation of their doubly-even magic torus descendants. In addition, a fourth-order Candy-style perfect square pandiagonal torus, and a fourth-order unidirectionally pandiagonal semi-magic torus, are each analysed in detail, and their torus descendants generated, throughout their respective higher-orders.

At the end of the study observations are made, and some conclusions are drawn as to the signification of the findings, and the potential for future research.

Some sample illustrations of a fourth-order partially pandiagonal torus and its partially pandiagonal torus descendants are shown below:

T4_195_magic_torus
T4.195 4th-order partially pandiagonal torus

T4_195_magic_torus_order_8_descendant
8th-order partially pandiagonal torus, direct descendant of T4.195

T4_195_magic_torus_order_12_descendant
12th-order partially pandiagonal torus, direct descendant of T4.195

T4_195_magic_torus_order_16_descendant
16th-order partially pandiagonal torus, direct descendant of T4.195

To download "Magic torus coordinate and vector symmetries," please use the following link: MTCVS 160910.pdf. The 134M pdf file exceeds the maximum size (25M) that can be virus scan checked by Google Drive, but as the file is virus free you can ignore the warning and safely download the document.