Archive for the ‘Unusual Geometry’ Category

Isohedral Tiling

May 6, 2017

I don’t recall anymore what path I was wandering when I came across this, but it falls in the ‘different tilings’ category which seems to be an onging interest of mine.  It’s really a 3 piece pattern of a hex grid.

Link: https://en.wikipedia.org/wiki/File:Isohedral_tiling_p6-11.png

And here’s my cleaner version of this tiling.  With more of pattern visible, you can more readily see the hex pattern made up of these triplets.

It’s not really anything that’s going to be transformative, but it may be useful as a different way of looking at a grid and of having a different graphic presentation.

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Voronoi Map – Settlement and Wilderness

September 15, 2015

There’s not a whole lot of progress on the Voronoi map I started a while ago, but this was an interesting exercise, and maybe worth sharing.

CaptureVoronoiField

This is the same as the previous Voronoi map, but with some new information added.  Each cell of the map that had a village, as well as each town cell and all the cells immediately adjacent to it, were designated as ‘Inhabited,’ and are shaded blue on this map.  Then a buffer of one cell, adjacent to an inhabited cell, was chosen, and anything beyond there (indicated by the red lines on the map) is wilderness.

The rivers (which, for the most part, I’ve chosen to make follow the edges between cells) also act as the buffer, which is why the inhabited area at the bottom center is apparently directly adjacent to a wilderness area, because it’s across the river.

This makes some interesting effects, in terms of where wildness comes surprisingly close to civilization.  I’m also intrigued by how much of the river in the center of the map is buffer region, but then it switches to wilderness.

To me, this starts to set up some character of the region; it suggests where farms might be, and even starts to suggest a couple places that are ‘civilized,’ but where danger might be higher because the wilderness is not that far away.

A Voronoi Map (part 1)

August 26, 2015

voronoiCapture After the pentagonal tiling map was posted last week, there were a couple comments that discussed Voronoi diagrams as an alternative.  This illustration is an example.

It’s another kind of grid-making, although it doesn’t produce regular sized regions, like a tiling pattern does. Nonetheless, it could be used as another way of generating walls with irregular edges, good for caverns and caves, though my own preference would be to have something regular for an interior space such as a dungeon.

But a Voronoi diagram might be a good way of making a regional map, with different kinds of terrain.  So this is a documentation of the experiment.  I’m not sure where it’s headed, but I’ll post updates to it as it progresses.

IMAG1097 The first steps were making villages (where two points were close to one another) and towns (where there was a cluster with a center).  Town/cities are not just the center of the cluster, but also extend to all other immediately adjacent cells in the diagram.  Some coastline was also added at the top, as well.  This is all somewhat arbitrary, and I’m fairly sure I’m not following the rules with rigor.  But it does provide a way to make decisions, and it suggests differences in the landscape, unlike a blank page or any sort of regular grid would.

Next are some rivers, and a bit of shading to show the coastline more clearly.  Rivers always follow the edges, since they define a separation between two areas (except for the delta wetland of the left river at the coast, which becomes the whole area.  Areas around the towns were also shaded to highlight them a little more.

IMAG1098Roads sometimes follow edges, and serve as the boundary between two regions, but at other times, they go through the middle of a region.  Again, it’s arbitrary choices (and the beginnings of stories about places, potentially). A road going between two areas serves to define the division between them.  But a road that goes through the center of a region serves the region and is part of its character.

This is all rough diagramming at this point.  The final version, even if it is drawn on a copy of the Voronoi diagram, will start with a clean copy of the diagram.  But the rough version is fine for working things out.

IMAG1101

The most recent version has added some more roads, as well as adding some features.  Areas with a small region are likely a point of interest of one sort or another, so a couple of those have been marked with diamonds.  Some forested areas of woodland and some mountainous areas have been identified, as well.

Pentagonal Grid Sheet for Sharing

August 20, 2015

A lot of people were really interested in the Pentagonal Tiling Map posted earlier this week.  And a couple people have asked for a clean copy of the grid I used to make that map, so I have a JPG and a PDF version I’m making available.  I would be very interested in seeing any other examples anyone creates using these grids, and I’m especially interested in hearing about it if anyone actually uses the pentagon grid in a game.

type7tilingdiagramThat grid was made using the Type 7 tiling description (see illustration) from Convex Pentagons for Edge-to-Edge Tiling, I – a mathematical paper, but with useful diagrams of the then-known types of pentagonal tilings (14, at that time).  It’s a good resource for looking at the other types, in case you want to experiment with making your own.

For most, a JPG image is likely good enough for your purposes, and the image link at the bottom will probably be all that you need (click on it to go to the full size).  You’ll still have to scale it up to fit your paper size, but I think it should work reasonably well for that.  If you want the Letter size PDF file, you can download that, too.

(Please note, this was a quick-and-dirty version I just banged out to give the idea a try. There was no cleanup, and I see a few errors that would need to be fixed before a final version, but this should be perfectly adequate for experimentation.)

I’m planning to explore some of the other tilings, so there may be better resources in the future, with a collection of more patterns to use.

pentagon7tilinggrid

Pentagonal Tiling Map

August 18, 2015

This past weekend, there was an interesting article I came across with a really interesting graphic.  The article was about a group of mathematicians who discovered a new kind of pentagon that will tile the plane.  Gamers are well acquainted with grids of regular polygons that tile the plane, squares and hexes being familiar to us all.

map816-pent

According to the article, there are now 15 different pentagonal tilings that can completely cover a plane.  (There are also Penrose tiles, something else that I am very intrigued by, but there are some differences with those.  The most fascinating thing is that it is an irregular tiling of the plane.  But also, it takes two different tiles, “kites” and “darts” (two different parallelograms) for the tiling, although interestingly, the patterns that are created in Penrose tiles are also pentagonally-based, with lots of 72 degree angles in the field.  But I digress.)

15pentagonal tilings

Some of these tilings are more suited to be a hatch pattern than a grid.  But others, like the one I’ve used, have a cell that is large enough to make a decent grid network, and it seemed worth exploring whether or not a dungeon could be drawn using one of those tilings as the grid.  And one of these patterns (top row, middle) is actually a hexagonal tiling, with each hex divided into three matching pentagons.

map816-pentgrayTo my mind, the experiment was a success.  There’s an irregular, organic quality to this map, but it is all straight lines (almost all the same length; 4 of the 5 sides of the pentagon in this tiling are equal) and a handful of repeated angles.

There’s also a version where I filled the dungeon floor with a light gray.  (I thought it might help increase the readability of the map, but it may not be everyone’s preference.

I’m not sure how it would play out to use this as the dungeon map for a game (but if you give it a try, I would love to hear about it).  You could certianly overlay a conventional grid over the top of this, and use it much like you’d use any other map with irregular walls that don’t conform to the grid.  But I think it would be even better to try it out with the pentagonal grid as the game grid.  (For facing, you could have one front facing side of the pentagon, two flanking sides, left and right, and then just two rear sides.)  Certainly some things might get awkward in places, but I think it would be a lot of fun.  You could use this as a map of a dungeon withing some powerful magical vortex, or on an alternate plane, where Euclidian geometry doesn’t work, and everything the players (and the characters) know is not quite what they thought it was.

I’ve got larger, high resolution versions of both of these; those will be Patreon offerings.

These maps are posted here to share, and as is typical for these, you can use it for any personal use, but please only share them with attribution {CC-BY-NC 2015 by Rodger Thorm}  If you have a project in mind that you’d like to use this (or one of my other maps) for, drop me a line.

[image credit for pentagonal tilings: Ed Pegg Jr/Wikimedia Commons via NPR]