Update: Ponting packing/sequence grids & tilings

Thanks largely to very kind motivation/interest from Ed Pegg and Stuart Anderson, in 2017 I’ve done more stuff to do with “Ponting packings/patches” and sequence grids/tilings:

I wrote a paper on sequence grids & tilings : square tilings paper.pdf

pontingpacking.pyx : Cython program that prints sizes & locations (on lower left corner) of squares, and saves png image. See bottom of this page.

The width and height of the N\times N Ponting packing are the same, and given by \frac{N}{4}(3N^2-2N+3).
Their shape as N increases quickly gets close to the same warped square shape:




A few sequence grid tilings:
\textHVline{-1}{5}{}{0}{9}{} :

\textHVline{-1}{2}{}{0}{3}{} :

\textHVline{0}{1}{}{0}{2}{} :


I made various sequence grid tiling programs/pictures in 2017, but haven’t put them online yet, e.g.

3D version of \textHVline{3}{1}{}{0}{5}{}

Sequence grid circle inversions, using the square sizes and locations.
Detail of 31×31 \textHVline{26}{6}{}{23}{28}{} :

Detail of 31×31 \textHVline{3}{1}{}{0}{5}{} :

Detail of 31×31 \textHVline{50}{0}{}{47}{52}{} :

sequence tiling Voronoi art – the points are at the centres of sequence grid tiling squares.


#pontingpacking.pyx - v1 - 7 Oct 2017
#prints square sizes and coordinates and makes png image of packing
#NB the coordinates are with y=0 at the top. Subtract each y value from
#N*(3*N*N-2*N+3)/4+2*MARGIN for normal Cartesian coords.
#type OK() to run
import numpy as np
cimport numpy as np
from scipy import misc
DEF N = 15 #makes NxN packing
cdef struct pt:
	int x,y
cdef struct sqinfo:
	pt loc 
	int size
cdef sqinfo Square[N][N]
fn = '/Users/admin/%dx%dpacking.png' % (N,N)
cdef int highest

cdef GetSizes():
	cdef int i,j,count = 1
	#do bottom row
	for i in xrange(N-1,-1,-2): # in C: for (i=N-1;i>-1;i=i-2) {
		Square[i][N-1].size = count
		count += 1
	#rest of rows in pairs
	for j in xrange(N-2,0,-2):
		for i in xrange(N-2,0,-2):
			Square[i][j].size = count
			count += 1
		for i in xrange(N-1,-1,-2):
			Square[i][j-1].size = count
			count += 1
	#do right column
	for j in xrange(1,N-1,2): 
		Square[N-1][j].size = count
		count += 1
	#rest of columns in pairs
	for i in xrange(N-2,0,-2):
		for j in xrange(0,N,2):
			Square[i][j].size = count
			count += 1
		for j in xrange(1,N,2):
			Square[i-1][j].size = count
			count += 1
cdef GetPositions():
	global highest
	cdef int i,j
	cdef pt startOfRow,currentPos
	#do top row from left to right
	Square[0][0].loc = [0,0] #i.e. bottom left corner of top left square is at 0,0
	startOfRow = [0,0]
	currentPos = [0,0]
	for i in xrange(1,N-1,2):
		currentPos.x += Square[i-1][0].size
		Square[i][0].loc = currentPos
		currentPos.x += Square[i][0].size
		currentPos.y -= Square[i][0].size-Square[i+1][0].size
		Square[i+1][0].loc = currentPos
	#store highest point, i.e. smallest y value in packing
	#which is top of square at right end of top row
	highest = currentPos.y-Square[i+1][0].size
	#do rest of rows in pairs
	for j in xrange(1,N-1,2):
		startOfRow.y += Square[0][j].size
		Square[0][j].loc = startOfRow
		currentPos = startOfRow
		for i in xrange(1,N-1,2):
			currentPos.x += Square[i-1][j].size
			currentPos.y -= Square[i-1][j].size-Square[i][j].size
			Square[i][j].loc = currentPos
			currentPos.x += Square[i][j].size
			Square[i+1][j].loc = currentPos
		startOfRow.x += Square[0][j].size-Square[0][j+1].size
		startOfRow.y += Square[0][j+1].size
		Square[0][j+1].loc = startOfRow
		currentPos = startOfRow
		for i in xrange(1,N-1,2):
			currentPos.x += Square[i-1][j+1].size
			Square[i][j+1].loc = currentPos
			currentPos.x += Square[i][j+1].size
			currentPos.y -= Square[i][j+1].size-Square[i+1][j+1].size
			Square[i+1][j+1].loc = currentPos

cdef CorrectCoords():
	cdef int i,j
	for j in xrange(N):
		for i in xrange(N):
			Square[i][j].loc.y -= highest-MARGIN #so none are <MARGIN
			Square[i][j].loc.x += MARGIN

cdef PrintSizesAndLocations():
	cdef int i,j
	for j in xrange(N):
		for i in xrange(N):
			print "%3d" % Square[i][j].size,
	print "Locations of bottom left corners of squares:"
	for j in xrange(N):
		print "Row",j,":",
		for i in xrange(N):
			print "(%d,%d)" % (Square[i][j].loc.x,Square[i][j].loc.y),

cpdef OK():
	cdef int width,i,j,m
	width = N*(3*N*N-2*N+3)/4
	cdef int IMWIDTH = width+2*MARGIN
	cdef np.ndarray[np.uint8_t,ndim = 3] pic = np.zeros((IMWIDTH,IMWIDTH,3),dtype = np.uint8)
	print N,"x",N," packing, dimensions =",width,"x",width
	for j in xrange(N):
		for i in xrange(N):
			for m in xrange(Square[i][j].size):
				pic[Square[i][j].loc.y,Square[i][j].loc.x+m,:] = 255
				pic[Square[i][j].loc.y-m,Square[i][j].loc.x,:] = 255
				pic[Square[i][j].loc.y-Square[i][j].size,Square[i][j].loc.x+m,:] = 255
				pic[Square[i][j].loc.y-m,Square[i][j].loc.x+Square[i][j].size,:] = 255

9×9 :

15×15 :

19×19 :