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Forum index » Science and Technology » Math » Recreational
paving with bricks of different sizes
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Patrick Hamlyn
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Joined: 03 May 2005
Posts: 45

PostPosted: Tue Jul 19, 2005 5:28 am    Post subject: Re: paving with bricks of different sizes Reply with quote

Bingo. Fault-free, more 10x10s, in fact a relatively even mix:
11111122222222233333333333334444445555555
ie 6 1's, 9 2's, 13 3's, 6 4's, 7 5's.
Area is 25x21, ie 125 * 105cm^2.

1122233332224444222555555
1122233332224444222555555
3335555553334444222555555
3335555553332223333555555
3335555553332223333114444
3335555553332223333114444
1122233332223335555554444
1122233332223335555554444
3322233332223335555551133
3344445555553335555551133
3344445555554444333444433
5544445555554444333444455
5544445555554444333444455
5522233332224444333444455
5522233332221155555533355
5522233332221155555533355
5533355555533355555533355
4433355555533355555533344
4433355555533311333322244
4433355555533311333322244
4422233332224444333322244

Here it is 'unwrapped'. Because it unwraps so neatly, you should be able to lay
the tiles in 105cm strips quite quickly and easily.

22233332224444
1122233332224444222555555
1122233332224444222555555
3335555553334444222555555
3335555553332223333555555
3335555553332223333114444
3335555553332223333114444
1122233332223335555554444
1122233332223335555554444
2223333222333555555113333
4444555555333555555113333
4444555555444433344443333
4444555555444433344445555
4444555555444433344445555
2223333222444433344445555
2223333222115555553335555
2223333222115555553335555
3335555553335555553335555
3335555553335555553334444
3335555553331133332224444
3335555553331133332224444
33332224444

For edging, you could get a few bricks 10cm by 15, 20, 30cm and probably end up
with no like sizes touching still.

I could email a GIF of it to you if you wanted.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
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Odysseus
science forum addict


Joined: 05 Jun 2005
Posts: 60

PostPosted: Tue Jul 19, 2005 4:28 am    Post subject: Re: paving with bricks of different sizes Reply with quote

Patrick Hamlyn wrote:
Quote:

snip

Of course you will want to edge the paving. If you know the exact size, I can
(possibly) make a tiling which is nicely bordered so you don't need to cut tiles
on the edge.


Maybe the 10x10s could find some use there.


--
Odysseus
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Patrick Hamlyn
science forum beginner


Joined: 03 May 2005
Posts: 45

PostPosted: Mon Jul 18, 2005 11:49 pm    Post subject: Re: paving with bricks of different sizes Reply with quote

Patrick Hamlyn <path@multipro.N_OcomSP_AM.au> wrote:


Quote:
The 10x10 tile, being the only one with a side of length 10, is apparently
'incompatible' with the others when like sizes may not touch. I can't find any
prototiles which include this tile. I'll search further (up to 20x20 so far) but
I suspect it's impossible.

I was too suspicious.

Here's a 24x24 but it has a vertical fault line and only one 10x10. Bigger
prototiles may be able to remedy both of these, I'll keep searching.

this is quite big on the ground, 120cm (I presume it's cm, since you're posting
from Belgium), which is a nice size.

Of course you will want to edge the paving. If you know the exact size, I can
(possibly) make a tiling which is nicely bordered so you don't need to cut tiles
on the edge.

Tiles:
12222222222223333333333334444444455555555

Fault line marked with |
|
115555333322233332225555
115555333322233332225555
222444455555533332225555
222444455555544443335555
222444455555544443335555
333444455555544443335555
333222333322244443334444
333222333322255552224444
333222333322255552224444
555555222444455552224444
555555222444455553333222
555555222444455553333222
555555333444455553333222
333222333555544445555553
333222333555544445555553
333222333555544445555553
444333222555544445555554
444333222555522233332224
444333222555522233332224
444333444433322233332224
335555444433355555544443
335555444433355555544443
335555444433355555544443
335555333322255555544443
|
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
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Patrick Hamlyn
science forum beginner


Joined: 03 May 2005
Posts: 45

PostPosted: Mon Jul 18, 2005 12:15 pm    Post subject: Re: paving with bricks of different sizes Reply with quote

puit <nospamplease@hotmail.com> wrote:

Quote:
These are the sizes that I ordered:
10x10
15x15
20x15
20x20
30x20
They will be delivered around 27th of July.
I guess that there will be twice as much square bricks as rectangular
ones. To account for the uneven proportions we could consider for
example 2 bricks of 10x10 as 1 brick of size 20x10.
Figures that wrap in 2 dimensions are indeed more aesthetic.

Thanks in advance,

Houston we have a problem.

The 10x10 tile, being the only one with a side of length 10, is apparently
'incompatible' with the others when like sizes may not touch. I can't find any
prototiles which include this tile. I'll search further (up to 20x20 so far) but
I suspect it's impossible.

Either you need to include another tile with a side of 10, or exclude this one.

For the rest of this post, I'll exclude the 10x10 and divide all lengths by 5,
so we're dealing with the other four tiles, labelled as follows
2: 3x3
3: 4x3
4: 4x4
5: 4x6

Furthermore disallowing touching of like sizes at corners eliminates nearly all
the interesting tilings, so I allowed it.

Searching only up to a maximum height/width of 21 (and minimum of 5), prototiles
can be made which use the following sets of tiles:
222222222222333333333333444444555
22222222222233333333334444445555
2222222222223333333344444455555
222222222222333333444444555555
222222222333333333444444555
222222222333333444555
22222222333333334444445555
2222222233333333444455
222222223333334444555
22222222333344445555
222222333333444455
22222233333344445555
22222233333444555
22222233334455
22223333445
2222334455
2223333344445555
222333445
2223334455
2223345

To distinguish 'wrapped' tilings from 'non-wrapped' we define the term 'fault
line'. This is a line, either horizontal or vertical, which can bisect the
prototile (and hence the entire tesselation) without cutting through an
individual tile. A 'fault-free' prototile is obviously more desirable. This set
of pieces, having such a small range of edge sizes, offers relatively few
tilings without fault lines. If you add some more tiles you will get more
interesting tilings. My program doesn't distinguish fault-free tilings, so I'm
not sure there aren't any. Several thousand tilings is too many to examine.

Here are prototiles for a few of these sets:

222222222333333333444444555
(vertical fault line, marked with |)
|
222555522255554444333
222555522255554444333
222555522255554444333
333555533355553333222
333555533355553333222
333555533355553333222
333444433344442224444
222444422244442224444
222444422244442224444
222444422244443334444
333322233332223335555
333322233332223335555
333322233332223335555
444433344443332225555
444433344443332225555
444433344443332225555
444433344443334444333
|

222222222222333333333333444444555
(horizontal fault line, marked with _)
222333222333355553333
222333222333355553333
_222333222333355553333_
554444555544443335555
554444555544443335555
554444555544443335555
554444555544443335555
223333555533332223332
223333555533332223332
223333444433332223332
555222444422244443335
555222444422244442225
555222444422244442225
555333322233344442225
555333322233322233335
555333322233322233335
444222444433322233334
444222444422255552224
444222444422255552224
444333444422255552224

2223334455
(two fault lines, ie the edges)
2224444555555
2224444555555
2224444555555
3334444555555
3332223334444
3332223334444
3332223334444
5555553334444
5555552223333
5555552223333
5555552223333

2223345
(three fault lines)
2224444555555
2224444555555
2224444555555
3334444555555
3332223333222
3332223333222
3332223333222

--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
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puit
science forum beginner


Joined: 13 Jun 2005
Posts: 8

PostPosted: Mon Jul 18, 2005 4:10 am    Post subject: Re: paving with bricks of different sizes Reply with quote

These are the sizes that I ordered:
10x10
15x15
20x15
20x20
30x20
They will be delivered around 27th of July.
I guess that there will be twice as much square bricks as rectangular
ones. To account for the uneven proportions we could consider for
example 2 bricks of 10x10 as 1 brick of size 20x10.
Figures that wrap in 2 dimensions are indeed more aesthetic.

Thanks in advance,
puit


Patrick Hamlyn wrote:
Quote:
puit <nospamplease@hotmail.com> wrote:


I want to pave a terrace with bricks of different sizes (for example
sizes 2x2, 2x3, 3x3 and 2x4; I do not know yet in what proportions the
bricks of different sizes are distributed). It should look as if the
bricks are randomly placed, but it is of course important that I can lay
the bricks quickly. So I have to prepare the job.

Is there any website with pictures or a program that generates images
with patterns of squares and rectangles that can fill the plain?


I have a program that can make up 'prototiles' for tiling the plane with various
rules. eg here is a prototile which has the rule that no like sizes may touch,
using the sizes you gave:

1333322331
4422444334
4422444334
4422444224
3331133223
3331133223
2244433444
2244433444
2244422444
1333322331

The edges all wrap. Here it is 'unwrapped':
22
3333223311
3333223311
4442244433
4442244433
4442244422
3333113322
3333113322
2244433444
2244433444
22444 444

You should be able to put these together pretty quickly in rows. I can also tell
it to wrap only in dimension, so you get something that can be tiled in fixed
rows with no overlap between rows, but that wouldn't look as nice.

You could use just square tiles of various sizes, with a somewhat bigger
prototile. Bigger prototiles would make it a bit harder to keep track of the
pattern while laying, but it would be harder for someone to make out the
repeating pattern.

eg this is the smallest prototile with just square tiles, no like sizes
touching, even at a corner:
4444
33322 4444
33322 4444
33366663334444
66663332255555
66663332255555
66661444455555
22333444455555
22333444455555
44443334444333
44445555522333
44445555522333
444455555 1
55555
55555

Let me know what tiles you want to use, I'll see if I can make a prototile that
suits.
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Patrick Hamlyn
science forum beginner


Joined: 03 May 2005
Posts: 45

PostPosted: Thu Jul 14, 2005 5:13 pm    Post subject: Re: paving with bricks of different sizes Reply with quote

puit <nospamplease@hotmail.com> wrote:

Quote:
I want to pave a terrace with bricks of different sizes (for example
sizes 2x2, 2x3, 3x3 and 2x4; I do not know yet in what proportions the
bricks of different sizes are distributed). It should look as if the
bricks are randomly placed, but it is of course important that I can lay
the bricks quickly. So I have to prepare the job.

Is there any website with pictures or a program that generates images
with patterns of squares and rectangles that can fill the plain?

I have a program that can make up 'prototiles' for tiling the plane with various
rules. eg here is a prototile which has the rule that no like sizes may touch,
using the sizes you gave:

1333322331
4422444334
4422444334
4422444224
3331133223
3331133223
2244433444
2244433444
2244422444
1333322331

The edges all wrap. Here it is 'unwrapped':
22
3333223311
3333223311
4442244433
4442244433
4442244422
3333113322
3333113322
2244433444
2244433444
22444 444

You should be able to put these together pretty quickly in rows. I can also tell
it to wrap only in dimension, so you get something that can be tiled in fixed
rows with no overlap between rows, but that wouldn't look as nice.

You could use just square tiles of various sizes, with a somewhat bigger
prototile. Bigger prototiles would make it a bit harder to keep track of the
pattern while laying, but it would be harder for someone to make out the
repeating pattern.

eg this is the smallest prototile with just square tiles, no like sizes
touching, even at a corner:
4444
33322 4444
33322 4444
33366663334444
66663332255555
66663332255555
66661444455555
22333444455555
22333444455555
44443334444333
44445555522333
44445555522333
444455555 1
55555
55555

Let me know what tiles you want to use, I'll see if I can make a prototile that
suits.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
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KBH
science forum beginner


Joined: 23 May 2005
Posts: 35

PostPosted: Thu Jul 14, 2005 3:06 pm    Post subject: Re: paving with bricks of different sizes Reply with quote

Quote:
Is there any website with pictures or a program that generates images with
patterns of squares and rectangles that can fill the plain?


'Scratch-Plot' allows drawing of rectangles probably most convenient when in
Azimuth-Distance mode.

Here is a user link to 'Scratch-Plot':

http://www.kbhscape.com/plot.htm
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puit
science forum beginner


Joined: 13 Jun 2005
Posts: 8

PostPosted: Thu Jul 14, 2005 12:48 pm    Post subject: paving with bricks of different sizes Reply with quote

I want to pave a terrace with bricks of different sizes (for example
sizes 2x2, 2x3, 3x3 and 2x4; I do not know yet in what proportions the
bricks of different sizes are distributed). It should look as if the
bricks are randomly placed, but it is of course important that I can lay
the bricks quickly. So I have to prepare the job.

Is there any website with pictures or a program that generates images
with patterns of squares and rectangles that can fill the plain?

Thanks,
puit
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