Thanks to Ed Pegg for the great idea of using 2 numbers for each shape, for height and width. (How extremely stupid not to have thought of that!)
I wrote a rectangle tiling program in Cython, which produces output like this:
If squares are input, the tilings are the same as the Double Rainbow family of square tesselations.
The only problem so far is that, unlike with squares, the rectangles with negative sides don’t fit into the tiling. Maybe the ones with both sides negative do; I haven’t even tried that yet. I suspect there’s some way of getting it all to work.
But for now, the program prints only the rectangles with positive height and width.
Rectiling.pyx (Sage cython) – save this file as Rectiling.pyx
To run on the Sage command line, “%attach Rectiling.pyx”. To run in a Sage notebook, paste into a cell, uncommenting the first line.
The user-edited variables I’ve put in 2 places between “——” lines.
The program first calculates the h-/v-line grid (and optionally produces a png of that), then calculates the tiling.
The rectangles are input (near the bottom of the program) with the EnterSize function, in the format ([xpos,ypos],width,height). It seems that it needs 6 rectangles specified to make a tiling, preferably mostly in a block together, so there are arithmetic sequences to extend. One of them must be at (0,0), the centre of the grid. If not enough rectangles are input, the grid will be partially empty, and further rectangles can be entered as needed to make the tiling.
The initial rectangles included here (used in the above pic) –
EnterSize([0,1],10,20) EnterSize([1,1],12,16) EnterSize([0,0],9,14) EnterSize([1,0],13,13) EnterSize([0,-1],8,9) EnterSize([1,-1],3,10),
– I chose because it has a lot of different sequences in it:
Widths: increase by 3 and 5 towards the upper right, by 6 and -3 to the upper left.
Heights: increase by 2 and 4 to the upper right, by 4 and 7 to the upper left.
It seems there can be up to 6 different sequence increments in a single rectiling, here 2,3,4,5,6 and 7.
NB The input squares must form a h-line with the upper 4 widths, and a v-line with the lower 4 heights. i.e. in the above rectangle data, these sums must be equal:
Top 4 rectangle widths: 10+12=9+13 (h-line)
Bottom 4 rectangle heights: 14+9=13+10 (v-line)
Apart from that restriction, the values can be any integers at all.
e.g. the 20,16,8 and 3 (the upper 2 heights and lower 2 widths) can be each changed separately and arbitrarily, not being part of the initial h-line or v-line. e.g. just change the 10×20 to a 10×15,10,3,-30 or anything, and see the tiling change:
A detail from (10,0):
From (10,-20):
From (10,30):
22-9-12-9-9-4-25-7-19-6-5-3
22-9-12-9-9-4-25-7-24-6-5-3
22-9-12-9-9-4-25-7-24-6-5-3
1-8-1-4-0-1-2-2-2-0-5-1
1-4-1-4-0-1-2-2-2-0-5-1
1-2-1-4-0-1-2-2-2-0-5-1
More pics here soon..
Rectiling.pyx:
#%cython #uncomment this line to run in a Sage notebook
# Rectiling v1 - 8 Jan 2017
#
#Improvements to make
#- put bool option, whether to print rects with neg edges or not.
#- add appropriate colouring for square tilings.
#- check if input/enter vals add up properly. ERROR if not.
from sage.misc.functional import show
from sage.plot.graphics import Graphics
from sage.plot.line import line2d
from sage.plot.text import text
from sage.plot.polygon import polygon2d as p
#NB polygon edgecolor has only worked since 2014;download the new polygon.py if needed
from sage.misc.latex import latex
from libc.math cimport abs
#--------------------------------------------------
#USER-EDITED VALUES
DEF cx=30*4 #x-pos of central rect in grid - NB keep as a multiple of 4.
#Change cx here to affect number of rects calculated: approx (2*cx)^2
cdef bint PrintGrid=False #also makes png of the grid if True
DEF GRIDPICSIZE=12 #size of grid png produced
cdef bint LabelTiling=False # writes sizes on tiling if True
DEF TILPICSIZE=16 #size of tiling png produced
DEF EDGE=0.5 #black edge width, try 0.5 or 1
cdef bint ColourIn=True #make coloured tiling if True
DEF MAXSIDE=250.0 #for colouring - set to approx largest side length
#MAXSIDE controls how big the rects have to be before the red level maxes out
#i.e. a smaller value will make the tiling look redder.
#Current colour map: skinnier=greener. shorter=bluer. bigger=redder
#--------------------------------------------------
DEF cy=cx #y-pos in grid of initial rectangle.
DEF MAXDIM=2*cx
DEF gridwidth=cx-4 # number of cells x cells to show on grid. must be <cx,cy
DEF H=gridwidth/2 # <-- NB must be even!! won't work otherwise (so cx must be 4n)
DEF TILH=H-1 #half-width of tiling to draw
cdef:
bint InfoAdded
int i,j,k
struct r:
int w,h #width,height of rectangle
bint WK,HK #true if width, height known
int L,R,T,B #x/y loc of sides in tiling
bint Drawn #true if drawn in tiling already
r g[MAXDIM][MAXDIM], *C
cdef EnterSize(int x[2], int wid, int hei): #enter a few rects to start with. See below
C = &g[x[0]+cx][x[1]+cy]
C.w=wid
C.h=hei
C.WK=True
C.HK=True
cdef ExtrapolateWidths(int xoff, int yoff): #try filling widths diagonally from known squares
global InfoAdded
cdef r *opp=&g[i-xoff][j-yoff] #i.e. 'opp' is on the opposite side of current square to 'new'
cdef r *new=&g[i+xoff][j+yoff]
if opp.WK and not new.WK: #ie 2 in a row, fill the third one with sequence
new.w=(g[i][j].w-opp.w)+g[i][j].w
new.WK=True
InfoAdded=True
cdef ExtrapolateHeights(int xoff, int yoff):
global InfoAdded
cdef r *opp=&g[i-xoff][j-yoff] #i.e. 'opp' is on the opposite side of current square to 'new'
cdef r *new=&g[i+xoff][j+yoff]
if opp.HK and not new.HK:
new.h=(g[i][j].h-opp.h)+g[i][j].h
new.HK=True
InfoAdded=True
cdef CheckHline(int i, int j): #checks if 3 rects from an hline already known, if so adds the 4th
global InfoAdded
cdef r *a, *b, *c, *d
a=&g[i][j+1]
b=&g[i+1][j+1]
c=&g[i][j]
d=&g[i+1][j]
if c.WK and a.WK and b.WK and not d.WK: #if a,b,c known, but not d
d.w=a.w+b.w-c.w # d=a+b-c
d.WK=True
InfoAdded=True
if d.WK and a.WK and b.WK and not c.WK:
c.w=a.w+b.w-d.w
c.WK=True
InfoAdded=True
if a.WK and c.WK and d.WK and not b.WK:
b.w=c.w+d.w-a.w
b.WK=True
InfoAdded=True
if b.WK and c.WK and d.WK and not a.WK:
a.w=c.w+d.w-b.w
a.WK=True
InfoAdded=True
cdef CheckVline(int i, int j): #checks for 3 rects from a v-line
global InfoAdded
cdef r *a, *b, *c, *d
a=&g[i][j+1]
b=&g[i][j]
c=&g[i+1][j+1]
d=&g[i+1][j]
if c.HK and a.HK and b.HK and not d.HK:
d.h=a.h+b.h-c.h
d.HK=True
InfoAdded=True
if d.HK and a.HK and b.HK and not c.HK:
c.h=a.h+b.h-d.h
c.HK=True
InfoAdded=True
if a.HK and c.HK and d.HK and not b.HK:
b.h=c.h+d.h-a.h
b.HK=True
InfoAdded=True
if b.HK and c.HK and d.HK and not a.HK:
a.h=c.h+d.h-b.h
a.HK=True
InfoAdded=True
cdef bint Extrapolate():
global i, j, k, InfoAdded
cdef int u, v
InfoAdded=False
for i in range(cx-H,cx+H):
#tries to fill empty grid numbers. returns True if it's made progress, else False.
for j in range(cy-H,cy+H):
if g[i][j].WK:
for u in range(-1,2,2):
for v in range(-1,2,2):
ExtrapolateWidths(u,v)
if g[i][j].HK:
for u in range(-1,2,2): #i.e. -1 then 1
for v in range(-1,2,2):
ExtrapolateHeights(u,v) #try extending heights
for i in range(cx-H,cx+H+1): #now try to fill in h-lines.
for k in range(cy-H,cy+H+1,2): #cy-H must be even
j=k+i%2 #to get the checkerboard pattern. i & j are both odd or both even.
#let c (i+0,j+0) = bottom left of an h-line.
CheckHline(i,j)
for i in range(cx-H,cx+H+1): #try to complete v-lines
for k in range(cy-H+1,cy+H+1,2): #cy-H+1 must be odd
j=k+i%2 #i & j are now opposite parity.
CheckVline(i,j)
return InfoAdded # true if anything has been learnt/added this time through
cdef ShowGridVals(): #draws the known grid values on the grid
global f,i,j
for i in range(cx-H,cy+H):
for j in range(cx-H,cy+H):
C=&g[i][j]
if C.WK: #if width known, print it
f+=text("${}$".format(C.w),(i+0.25,j+0.75))
if C.HK:
f+=text("${}$".format(C.h),(i+0.75,j+0.25))
cdef int Colour(float dim):
dim=255-dim*(255.0/MAXSIDE)
if dim<0:
return 0
else:
return int(dim)
cdef Draw(int x,int y):
global f
cdef r *S = &g[x][y]
cdef int red
cdef float fw,fh
rect=[[S.L,S.B],[S.R,S.B],[S.R,S.T],[S.L,S.T]]
if ColourIn:
fw=float(abs(S.w))
fh=float(abs(S.h))
red=int(255.0*(fw*fh)/(MAXSIDE*MAXSIDE)) #bigger = redder
if red>255: red=255
if red<0: red=0
f+=p(rect,color='#%02x%02x%02x' % (red,Colour(fw),Colour(fh)), \
edgecolor="black",thickness=EDGE)
else:
f+=p(rect,fill=False,color="black")
if LabelTiling:
f+=text("${},{}$".format(S.w,S.h),((S.L+S.R)/2.0,(S.T+S.B)/2.0),color="black")
S.Drawn = True
cdef DrawType1AndFindNeighboursSides(int X, int Y):
if X>cx+TILH or X<cx-TILH or Y>cy+TILH or Y<cy-TILH:
return
cdef r *N, *S = &g[X][Y]
Draw(X,Y)
#given a type 1 rect at X,Y, work out neighbours' sides positions
N=&g[X+1][Y] #the rectangle to right
if not N.Drawn and N.h>0 and N.w>0: #if not drawn yet, and has sides>0
SetSides(N,S.T,S.L+N.w,S.T-N.h,S.R)
DrawType2AndFindNeighboursSides(X+1,Y)
N=&g[X][Y+1] #the rectangle above
if not N.Drawn and N.h>0 and N.w>0:
SetSides(N,S.T+N.h,S.L+N.w,S.T,S.L)
DrawType2AndFindNeighboursSides(X,Y+1)
N=&g[X][Y-1] #the rectangle below
if not N.Drawn and N.h>0 and N.w>0:
SetSides(N,S.B,S.R,S.B-N.h,S.R-N.w)
DrawType2AndFindNeighboursSides(X,Y-1)
N=&g[X-1][Y] #the rectangle to left
if not N.Drawn and N.h>0 and N.w>0:
SetSides(N,S.B+N.h,S.L,S.B,S.L-N.w)
DrawType2AndFindNeighboursSides(X-1,Y)
cdef DrawType2AndFindNeighboursSides(int X, int Y):
if X>cx+TILH or X<cx-TILH or Y>cy+TILH or Y<cy-TILH:
return
cdef r *N, *S = &g[X][Y]
Draw(X,Y)
#given type 2 rect at X,Y, work out neighbours' sides positions
N=&g[X+1][Y] #the rectangle to right
if not N.Drawn and N.h>0 and N.w>0:
SetSides(N,S.B+N.h,S.R+N.w,S.B,S.R)
DrawType1AndFindNeighboursSides(X+1,Y)
N=&g[X][Y+1] #the rectangle above
if not N.Drawn and N.h>0 and N.w>0:
SetSides(N,S.T+N.h,S.R,S.T,S.R-N.w)
DrawType1AndFindNeighboursSides(X,Y+1)
N=&g[X][Y-1] #the rectangle below
if not N.Drawn and N.h>0 and N.w>0:
SetSides(N,S.B,S.L+N.w,S.B-N.h,S.L)
DrawType1AndFindNeighboursSides(X,Y-1)
N=&g[X-1][Y] #the rectangle to left
if not N.Drawn and N.h>0 and N.w>0:
SetSides(N,S.T,S.L,S.T-N.h,S.L-N.w)
DrawType1AndFindNeighboursSides(X-1,Y)
cdef SetSides(r *C,int T, int R, int B, int L):
C.T=T
C.R=R
C.B=B
C.L=L
#Initialize vars
for i in range(MAXDIM):
for j in range(MAXDIM):
C=&g[i][j]
C.WK=False
C.HK=False
C.Drawn=False
SetSides(C,-1000,-1000,-1000,-1000) #mainly so theyre not >0
#--------------------------------------------------
#manually enter known vals
EnterSize([0,1],10,20)
EnterSize([1,1],12,16)
EnterSize([0,0],9,14)
EnterSize([1,0],13,13)
EnterSize([0,-1],8,9)
EnterSize([1,-1],3,10)
#e.g. to get a square tesselation
# EnterSize([0,1],5,5)
# EnterSize([1,1],2,2)
# EnterSize([0,0],3,3)
# EnterSize([1,0],4,4)
# EnterSize([0,-1],10,10)
# EnterSize([1,-1],9,9)
#--------------------------------------------------
#draw grid
if PrintGrid:
f=Graphics()
for i in range(cx-H,cx+H,2):
for j in range(cx-H,cx+H,2):
for k in range(2):
f+=line2d([[i+0.1+k,j+1+k],[i+1.9+k,j+1+k]],color="black")
f+=line2d([[i+k+1,j+1.1-k],[i+k+1,j+2.9-k]],color="black")
while Extrapolate():
pass
if PrintGrid:
ShowGridVals()
show(f, aspect_ratio=1,figsize=GRIDPICSIZE)
f=Graphics()
#Now draw tiling
C=&g[cx][cy] #put grid rect [cy][cy] with its bottom left at (0,0), then work outwards
SetSides(C,C.h,C.w,0,0) #pointer,top,right,bottom,left sides.
#g[cx,cy] is called type 1 - bottom left of an h-line. its perp neighbours are type2.
DrawType1AndFindNeighboursSides(cx,cy)
show(f,aspect_ratio=1,figsize=TILPICSIZE,axes=False)