# Symmetry 2021 **Sofia/online** [Dave Everitt](https://daveeveritt.org/art.html) with [Fania Raczinski](https://fania.eu) artist-programmers <small>(brief biographies after main slides)</small>

## Magic square basics <small>rows, columns, diagonals share a common sum. Orders 3-></small> 

## The magic line <small>the path connecting each number in sequence (various methods shown)</small> 

## First magic square software <small>Hypercard, visualiser for magic square collections, 1980s-90s</small> 

## Current magic square software <small>[Magic square explorer](https://squares.cubelife.org) 2019-now (Fania Raczinski + Dave Everitt) order 12 shown (JavaScript, SVG, Haskell, CouchDB)</small> 

## Magic squares: summary - **One** set of rules, **various** groups, **multiple** types - **Millions** of permutations from order 5+ <small>(e.g. order 5: **275,305,224**)</small> - Magic line can display both **symmetry/asymmetry** - Focus is magic squares of **natural numbers** running **sequentially** from 1..*n*² - Previous paper: [Creative Visualisation of Magic Squares](https://www.scienceopen.com/document?vid=ec68f646-d76e-4961-986e-1aa83123d146)

## The magic line length differing paths through the *n* x *n* grid/matrix <small>the **shortest** and **longest** possible magic line paths for order 4</small> 

## Identifying types (order 4) <small>Using the webapp to display how **existing magic square types overlap**:</small> 

## Existing groups <small>exisiting methods are **numerical** rather than visual or **pattern-oriented**</small> <small>e.g. row/column transpositions greate groups that may or may not share **magic line** patterns<br> 

## Common patterns <small> this identifies and isolates the **homologous magic lines**<br> the focus is on transformations the produce **identical magic line patterns** (rotated/reflected) </small> <small>**homologous**: “having a similar position, structure, value, or purpose”</small> 

## Magic line length **step 1:** find squares with the **same length** of **magic line** <small>the **path through the matrix** is the same **length**, but the **magic line pattern** may differ</small> <small>groups displaying **homologous magic lines** can have between 2-8 members</small> 

## Magic line homology **step 2:** group squares with a **common magic line pattern** <small>the **path through the matrix** is identical or **homologous** but may be **rotated** or **reflected**<br> this group has **8 members** </small> 

## Orient the magic lines **step 3:** rotate/reflect patterns to a **common orientation** <small> **rotate** and/or **reflect** the magic line and numbers<br> the “parent” square has the number 1 in the top left, by convention </small> 

## Identify transformations **step 4:** identify **numerical transformations** behind the common patterns to identify **three types**: - complementary (reverse) - *n*-addition - shift-reverse

## Complementary (reverse) <pre style="width: 22%;"> 1 2 3 4… 15 16 16 15 14 13… 2 1 <strong>-(n² + 1) <i>n</i> = order</strong> </pre> <small>this transformation is already recognised, but **both resulting magic squares** are **included** in **definitive sets** (e.g. the 880 of order 4), so the **magic line** is **duplicated**</small> Both sequences follow the **same path** through the *n* x *n* matrix 

## N-addition (shift) <pre style="width: 22%;"> 1 2 3 4… 15 16 5 6 7 8… 3 4 9 10 11 12… 7 8 13 14 15 16… 11 12 <strong> (+n, +n×2, +n×3)</strong> </pre> *n*-added numbers: **same path**, **same direction** 

## Shift-reverse <pre style="width: 22%;"> 1 2 3 4… 15 16 4 3 2 1… 6 5 8 7 6 5… 10 9 12 11 10 9… 14 13 <strong>(complementary+n, +n×2, +n×3)</strong> </pre> *n*-added numbers: **same path**, **reverse direction** 

## Homologous magic lines Can these eight magic squares, sharing the **same path**, be considered as a **group** with a **single parent**? 

# Summary - if we view magic squares as a **path** through a **square matrix** - this path can be traversed using **several numerical sequences** - some **groups** of sequences produce **homologous magic lines** - **order 4** magic square group members number **from 2 to 8**

# Proposal - use these common pattern families as **new groups** or **equivalence classes** (as the D₈ symmetries are now) - reduce the number of possible magic squares by only including the **parent** of the **magic line homology classes** - use the magic line homology groups to simplify calculations of **total magic squares** permutations

# Further work - how do the **existing group classifications** compare with the magic line **homologies?** - **which existing groups** (pandiagonal, associative, etc.) have<br>**1, 2, 4 or 8** members? - **higher order squares** also show homologous magic lines… - do higher order squares with homologous magic lines have **further numerical transformations**?

# Thank you! **Questions…** …and or a brief webapp **[basic demo](https://squares.cubelife.org/?order=4&amount=unique&style=numbers&size=8&gap=12&overlap=false&overlapAmount=overlap200&background=%23222222&stroke=%23FFFFFF&strokeWidth=2&salpha=255&fill=%23666666&falpha=0&animation=off&speed=50&dayMode=false)** (if time) or some **artistic magic square visualisations ->**

# Live demos - [Order 10 arc overlap](https://squares.cubelife.org/?order=10&amount=unique&style=arc&size=10&gap=15&overlap=true&overlapAmount=overlap200&background=%2337015b&stroke=%23ffef0a&strokeWidth=8&salpha=55&fill=%23ff9500&falpha=40&animation=sync&speed=100&dayMode=false&interface=hidden) - [Order 4 seamless grid animation quadline](https://squares.cubelife.org/?order=4&amount=unique&style=quadline&size=4&gap=0&overlap=false&overlapAmount=overlap200&background=%23222222&stroke=%23f8c8f9&strokeWidth=4&salpha=255&fill=%23666666&falpha=0&animation=async&speed=50&dayMode=false&interface=hidden) - [Order 5 straight](https://squares.cubelife.org/?order=5&amount=unique&style=straight&size=6&gap=0&overlap=false&overlapAmount=overlap200&background=%23222222&stroke=%23bcb8ff&strokeWidth=2&salpha=255&fill=%23666666&falpha=0&animation=async&speed=50&dayMode=false&interface=hidden) - [Order 12 pattern varieties (classes)](https://squares.cubelife.org/?order=12&amount=unique&style=straight&size=11&gap=26&overlap=false&overlapAmount=overlap200&background=%23222222&stroke=%23b8f9e9&strokeWidth=2&salpha=255&fill=%23666666&falpha=0&animation=off&speed=50&dayMode=false)

## About us **Dave**: - **Fine Art** > Loughborough **HCI** > [IoCT](https://www.ioct.co.uk/), De Montfort - 1980s: **magic square** Hypercard stack - 1999->: [**cubeLife**](https://cubelife.org) heartbeat-art (Loughborough Uni) - 2000s: [live art-code](https://e-artlab.com) (UK, Holland) - 2010s->: online works, [IoCT](https://www.ioct.co.uk/) De Montfort Uni **Fania**: - 2007: **Sudoku** solver (Computer Science, Leicester Uni) - 2010: live [**motion tracking**](https://vimeo.com/fania/) artwork, Masters IoCT - 2017: PhD [**NLP artwork + research**](https://pata.physics.wtf), [IoCT](https://www.ioct.co.uk/) - 200s-now: online works, [IoCT](https://www.ioct.co.uk/), De Montfort Uni