tiling mirroring and rotating. addition to surface mapping, part 1

Oh, i have a blog? seems I forgot it for a while :)  So some ideas on textures.

We use different textures for creating all kinds of materials for our 3D models.
In some cases we use tiling and mirroring for different ways to apply textures on existing 3D objects with uv coordinates.
But in some cases we use textures in architectural situations where we have, say, couple of tiles to cover whole wall. while having a nice texture of say one tile by tilling and mirroring we can,
make it cover whole wall.  To achieve some more variable results we use brick procedural map generator (in 3D max) which basically helps us to put our
tile image in more variable fashion.
But if you ever looked at islamic art and architectural tradition, you will see that some parts of ornaments and tilling can blow your mind with its complexity.

Here I add some articles, which are very interesting regarding this topick.

Geometry feat cloaked in medieval Islamic tile,
“Cintamani” and Islamic Tiles,
A Discovery in Architecture: 15th Century Islamic Architecture Presages 20th Century Mathematics,
Q1 Project – Islamic Tiles,
The Art of Mathematics Islamic patterns,
Sometimes you would look at the wall but fail to see contours of single pattern which is repeated.

tower decoration pattern

To some extent same can be said about baroque wallpapers in Europe. And probably in any culture and time period you could find very complicated pattern arrangements.

So while analysing couple of arabic patterns I thought why do i have to spend hours tilling and mirroring and rotating stuff in Photoshop, if it theoretically
could be done directly in material editor.
Many patterns have a basic triangle shape. So if we have correct texture which contains our desired basic pattern element in triangle shape, we could
instead of tilling or mirroring it rotate it. Sins all bitmaps, as far as i know, are rectangle in shape, we would need to find and identify our triangle element wich we
would use for rotation. technically I guess we could develop a type of bitmap which is a triangle in its shape. I guess its resolution would be defined not as
say 640×480 but i don’t know, 1 pixel at top row, 640 at bottom row, 480 pixels height or something like that.
But we will try to work with traditional bitmaps here.  So say we open a texture in our brand new material editor. then we should define a triangle in
one or another way. i guess our material editor should let draw a triangle on top of texture, something like guidelines in Photoshop. We would
make sure these guidelines matches our triangle shape in bitmap as close as possible. next step would be to chuse which corner of triangle is our pivot point,
or in other words where will be its rotation centre.

according to angles of our triangle new material editor should copy and rotate texture in a number of times which is closest to full circle.
(say our pivot point of triangle is 45 degrees, therefore material editor would copy/rotate our original image exactly 8 times, if i am still able to count :) )

So next thing, after we have nice circle with texture which repeats itself not by tiling or mirroring but by rotation, we would need to describe how to make many such
circles to cover our full wall with new pattern.
Here I discovered there are 3 types of triangles. perfect ones, good ones and bad ones.
perfect triangle is the one wich has 60 degrees angle in each corner (picture bee hive)
good one is when the degree of its pivotal angle is such number that after dividing 360 from that number u get something like: 1,2,3,4,5 .. 25…5356544444 – a whole number,
and not 5.22321445524… and this is also a description of bad triangle. what i mean is that the number to form a full circle should be always a whole number.

hexagon is formed out of 6 tryangles

anyways. so a triangle with 60 degrees corners can cover any surface. all other “good”triangles also cover whole surface, but with some gaps between themselves.