Modus Hopper Random: Good Plane Fun

Friday, 23 November 2018

Good Plane Fun

For this weekend, I have exactly seventeen unvisited step 4 grounds to choose from. This is not a number that lends itself easily to random methods involving dice, coins or sports events. However, I remember from my first year Crystallography course at Cambridge (yes, really) that there are exactly seventeen different ways of organising a repetitive pattern in two dimensions. As undergraduates, we learned about the two-dimensional case before moving on to the three dimensions needed for real crystal and mineral structures. Oh yes, life was good.

The seventeen plane symmetry groups have been known for a long time – the proof that seventeen was the full set was completed early in the 20^th century, but many ancient civilisations were using them in their decorative and artistic traditions.

Today, they are also known as the “wallpaper groups”. Keep reading folks, normal groundhopping will be resumed as soon as possible.

I have organised my matches into an alphabetical list, and linked each match to one of the plane symmetry groups.

Match	Plane Symmetry Group
Atherton Coll v Widnes	p1
Belper T v Loughborough Dyn	p2
Bideford v Moneyfields	pm
Brighouse T v Lincoln U	pg
Cleethorpes T v Stocksbridge PS	cm
Colne v Colwyn Bay	pmm
Felixstowe & Walton v Witham T	pmg
Glossop NE v Trafford	pgg
Market Drayton T v Skelmersdale U	cmm
Ossett Utd v Frickley Ath	p4
Pickering T v Ramsbottom U	p4m
Pontefract Coll v Carlton T	p4g
Prescot Cables v Clitheroe	p3
Runcorn Linnets v Newcastle T	p3m1
Sevenoaks v Guernsey	p31m
Slimbridge v Winchester C	p6
Street v Blackfield & Langley	p6m

As an example, the Persian tapestry shown above is in group cm and would send me to Cleethorpes. Patterns of the group cm have no elements of rotational symmetry, but they do have parallel mirror lines in one direction. The “unit cell”, the smallest repeating unit, is a rhombus – a quadrilateral with four sides of equal length but no internal angle of 90^o (which would make it either a rectangle or a square). Other groundhopping blogs are available.

You’re still here? Oh, good. However, this Persian glazed tile of group p4m would send me to Pickering Town. It has fourfold rotation symmetry – basically meaning that the pattern looks identical when you turn it through 90^o either way. You can do this four times in a complete rotation. There are also four “mirror lines”, horizontal, vertical and both diagonals. The “unit cell” is a square.

You get the idea. Now, here’s the thing. You are all surrounded by patterns – wallpapers, tiles, table surfaces, floors and so on. Anything that covers a flat surface.

The first person to tweet me a picture of a pattern from their home or workplace (as long as they don’t mind me then adding the photo to this blog) will decide my groundhopping destination. It is 11.20 am on Friday 23 November and I am about to publish. Update and news to follow soon!

UPDATE

We have a result! Thanks to @ThrstleFantstic for this picture of a cat feeding mat, annotated to show the symmetry elements of two perpendicular mirror lines (red) within a rhombic unit cell (yellow). There is considerable twofold rotational symmetry - meaning that if you turn the pattern through 180 degrees then it looks the same. The centres of rotational symmetry are the pale blue dots.

Hopefully, it can be seen that the symmetry elements of the cat mat are the same as in this piece of Persian tapestry.

... which is in plane group cmm ...

... which is why I will be at Market Drayton Town v Skelmersdale United tomorrow. Thank you for your interest - a #keepertopcolourstats result prediction (currently around 70% accurate) will appear on Twitter at 3.00pm.

Thanks too to @ianshillaker1 for this beauty, which arrived a few minutes later and as far as I can see belongs to group p1 and therefore gives me my reserve choice of Atherton Collieries.