Fractional oblongs in unit square via the Paulhus packing technique

Oblongs of size $ \frac{1}{1} \times \frac{1}{2}$, $ \frac{1}{2} \times \frac{1}{3}$, $ \frac{1}{3} \times \frac{1}{4}$, $ \frac{1}{4} \times \frac{1}{5}$, ... have a total area of 1.

$\sum\limits_{a=1}^\infty \frac{1}{a (a+1)} =1$

I believe it's still an unsolved question whether the infinite rectangles can fit in the unit square. If it's been solved, I'd love to see the paper.

There is a method by Paulhus for packing infinite fractional squares into a rectangle.

We can start with a rectangle with an area slightly greater than all the fractional squares. $$\sum\limits_{a=2}^\infty \frac{1}{a^2} = \frac{\pi^2}{6}-1 \approx 0.644934067 \approx 0.644934069 \approx \frac{545}{467} \times \frac{21}{38}$$

His coding method seems to work very well, as seen below.

However, I haven't been able to figure out quite how to modify the code to intelligently do oblongs instead of squares. Can anyone modify the code and get the first 2000 or so rectangles into the unit square? Or has that already been done?

1. M. M. Paulhus, "An Algorithm for Packing Squares," Journal of Combinatorial Theory, Series A, 82(2), 1998 pp. 147–157.

Solution 1:

Possible packing:

Here is scaled-up one ($2048\times 2048$ pixels).

And coordinates of first 200 rectangles ('h' - horizontal, 'v' - vertical):

1) h x1=0.000000000000 y1=0.000000000000 x2=1.000000000000 y2=0.500000000000
2) h x1=0.000000000000 y1=0.500000000000 x2=0.500000000000 y2=0.833333333333
3) h x1=0.500000000000 y1=0.500000000000 x2=0.833333333333 y2=0.750000000000
4) h x1=0.500000000000 y1=0.750000000000 x2=0.750000000000 y2=0.950000000000
5) h x1=0.000000000000 y1=0.833333333333 x2=0.200000000000 y2=1.000000000000
6) h x1=0.200000000000 y1=0.833333333333 x2=0.366666666667 y2=0.976190476190
7) v x1=0.366666666667 y1=0.833333333333 x2=0.491666666667 y2=0.976190476190
8) h x1=0.833333333333 y1=0.500000000000 x2=0.958333333333 y2=0.611111111111
9) h x1=0.833333333333 y1=0.611111111111 x2=0.944444444444 y2=0.711111111111
10) h x1=0.750000000000 y1=0.750000000000 x2=0.850000000000 y2=0.840909090909
11) h x1=0.750000000000 y1=0.840909090909 x2=0.840909090909 y2=0.924242424242
12) h x1=0.850000000000 y1=0.750000000000 x2=0.933333333333 y2=0.826923076923
13) h x1=0.750000000000 y1=0.924242424242 x2=0.826923076923 y2=0.995670995671
14) h x1=0.850000000000 y1=0.826923076923 x2=0.921428571429 y2=0.893589743590
15) h x1=0.933333333333 y1=0.750000000000 x2=1.000000000000 y2=0.812500000000
16) h x1=0.850000000000 y1=0.893589743590 x2=0.912500000000 y2=0.952413273002
17) v x1=0.944444444444 y1=0.611111111111 x2=1.000000000000 y2=0.669934640523
18) h x1=0.944444444444 y1=0.669934640523 x2=1.000000000000 y2=0.722566219470
19) h x1=0.500000000000 y1=0.950000000000 x2=0.552631578947 y2=1.000000000000
20) h x1=0.552631578947 y1=0.950000000000 x2=0.602631578947 y2=0.997619047619
21) h x1=0.602631578947 y1=0.950000000000 x2=0.650250626566 y2=0.995454545455
22) h x1=0.650250626566 y1=0.950000000000 x2=0.695705172021 y2=0.993478260870
23) v x1=0.958333333333 y1=0.500000000000 x2=1.000000000000 y2=0.543478260870
24) h x1=0.958333333333 y1=0.543478260870 x2=1.000000000000 y2=0.583478260870
25) h x1=0.833333333333 y1=0.711111111111 x2=0.873333333333 y2=0.749572649573
26) h x1=0.873333333333 y1=0.711111111111 x2=0.911794871795 y2=0.748148148148
27) h x1=0.695705172021 y1=0.950000000000 x2=0.732742209058 y2=0.985714285714
28) h x1=0.933333333333 y1=0.812500000000 x2=0.969047619048 y2=0.846982758621
29) h x1=0.933333333333 y1=0.846982758621 x2=0.967816091954 y2=0.880316091954
30) v x1=0.911794871795 y1=0.711111111111 x2=0.944052936311 y2=0.744444444444
31) h x1=0.850000000000 y1=0.952413273002 x2=0.882258064516 y2=0.983663273002
32) v x1=0.969047619048 y1=0.812500000000 x2=0.999350649351 y2=0.843750000000
33) h x1=0.912500000000 y1=0.893589743590 x2=0.942803030303 y2=0.923001508296
34) h x1=0.969047619048 y1=0.843750000000 x2=0.998459383754 y2=0.872321428571
35) h x1=0.882258064516 y1=0.952413273002 x2=0.910829493088 y2=0.980191050779
36) h x1=0.958333333333 y1=0.583478260870 x2=0.986111111111 y2=0.610505287897
37) h x1=0.944444444444 y1=0.722566219470 x2=0.971471471471 y2=0.748882008944
38) h x1=0.971471471471 y1=0.722566219470 x2=0.997787260945 y2=0.748207245111
39) h x1=0.912500000000 y1=0.923001508296 x2=0.938141025641 y2=0.948001508296
40) h x1=0.942803030303 y1=0.893589743590 x2=0.967803030303 y2=0.917979987492
41) h x1=0.200000000000 y1=0.976190476190 x2=0.224390243902 y2=1.000000000000
42) h x1=0.224390243902 y1=0.976190476190 x2=0.248199767712 y2=0.999446290144
43) h x1=0.248199767712 y1=0.976190476190 x2=0.271455581665 y2=0.998917748918
44) h x1=0.271455581665 y1=0.976190476190 x2=0.294182854393 y2=0.998412698413
45) h x1=0.294182854393 y1=0.976190476190 x2=0.316405076615 y2=0.997929606625
46) h x1=0.316405076615 y1=0.976190476190 x2=0.338144207050 y2=0.997467071935
47) h x1=0.338144207050 y1=0.976190476190 x2=0.359420802794 y2=0.997023809524
48) h x1=0.359420802794 y1=0.976190476190 x2=0.380254136128 y2=0.996598639456
49) h x1=0.380254136128 y1=0.976190476190 x2=0.400662299393 y2=0.996190476190
50) h x1=0.400662299393 y1=0.976190476190 x2=0.420662299393 y2=0.995798319328
51) h x1=0.420662299393 y1=0.976190476190 x2=0.440270142530 y2=0.995421245421
52) h x1=0.440270142530 y1=0.976190476190 x2=0.459500911761 y2=0.995058400719
53) h x1=0.459500911761 y1=0.976190476190 x2=0.478368836289 y2=0.994708994709
54) h x1=0.478368836289 y1=0.976190476190 x2=0.496887354808 y2=0.994372294372
55) h x1=0.826923076923 y1=0.924242424242 x2=0.845104895105 y2=0.942099567100
56) h x1=0.826923076923 y1=0.942099567100 x2=0.844780219780 y2=0.959643426749
57) v x1=0.732742209058 y1=0.950000000000 x2=0.749983588368 y2=0.967543859649
58) h x1=0.732742209058 y1=0.967543859649 x2=0.749983588368 y2=0.984493012191
59) h x1=0.826923076923 y1=0.959643426749 x2=0.843872229465 y2=0.976310093415
60) h x1=0.826923076923 y1=0.976310093415 x2=0.843589743590 y2=0.992703536038
61) h x1=0.850000000000 y1=0.983663273002 x2=0.866393442623 y2=0.999792305260
62) h x1=0.969047619048 y1=0.872321428571 x2=0.985176651306 y2=0.888194444444
63) h x1=0.866393442623 y1=0.983663273002 x2=0.882266458496 y2=0.999288273002
64) h x1=0.732742209058 y1=0.984493012191 x2=0.748367209058 y2=0.999877627576
65) h x1=0.969047619048 y1=0.888194444444 x2=0.984432234432 y2=0.903345959596
66) h x1=0.912500000000 y1=0.948001508296 x2=0.927651515152 y2=0.962926881430
67) v x1=0.985176651306 y1=0.872321428571 x2=0.999882533659 y2=0.887246801706
68) h x1=0.942803030303 y1=0.917979987492 x2=0.957508912656 y2=0.932472741115
69) h x1=0.695705172021 y1=0.985714285714 x2=0.710197925644 y2=1.000000000000
70) h x1=0.710197925644 y1=0.985714285714 x2=0.724483639930 y2=0.999798792757
71) v x1=0.986111111111 y1=0.583478260870 x2=1.000000000000 y2=0.597562767912
72) h x1=0.882266458496 y1=0.983663273002 x2=0.896155347385 y2=0.997361903138
73) h x1=0.986111111111 y1=0.597562767912 x2=0.999809741248 y2=0.611076281425
74) h x1=0.969047619048 y1=0.903345959596 x2=0.982561132561 y2=0.916679292929
75) h x1=0.933333333333 y1=0.880316091954 x2=0.946666666667 y2=0.893473986691
76) h x1=0.946666666667 y1=0.880316091954 x2=0.959824561404 y2=0.893303104941
77) h x1=0.985176651306 y1=0.887246801706 x2=0.998163664293 y2=0.900067314526
78) h x1=0.942803030303 y1=0.932472741115 x2=0.955623543124 y2=0.945130968963
79) h x1=0.912500000000 y1=0.962926881430 x2=0.925158227848 y2=0.975426881430
80) h x1=0.957508912656 y1=0.917979987492 x2=0.970008912656 y2=0.930325666505
81) h x1=0.927651515152 y1=0.948001508296 x2=0.939997194164 y2=0.960196630247
82) h x1=0.896155347385 y1=0.983663273002 x2=0.908350469336 y2=0.995711465773
83) v x1=0.921428571429 y1=0.826923076923 x2=0.933333333333 y2=0.838971269694
84) h x1=0.921428571429 y1=0.838971269694 x2=0.933333333333 y2=0.850735975577
85) h x1=0.921428571429 y1=0.850735975577 x2=0.933193277311 y2=0.862363882553
86) h x1=0.921428571429 y1=0.862363882553 x2=0.933056478405 y2=0.873858135427
87) h x1=0.921428571429 y1=0.873858135427 x2=0.932922824302 y2=0.885221771790
88) h x1=0.985176651306 y1=0.900067314526 x2=0.996540287669 y2=0.911303269582
89) h x1=0.957508912656 y1=0.930325666505 x2=0.968744867712 y2=0.941436777616
90) h x1=0.927651515152 y1=0.960196630247 x2=0.938762626263 y2=0.971185641236
91) h x1=0.912500000000 y1=0.975426881430 x2=0.923489010989 y2=0.986296446647
92) h x1=0.942803030303 y1=0.945130968963 x2=0.953672595520 y2=0.955883657136
93) h x1=0.970008912656 y1=0.917979987492 x2=0.980761600828 y2=0.928618285365
94) h x1=0.985176651306 y1=0.911303269582 x2=0.995814949178 y2=0.921829585372
95) h x1=0.970008912656 y1=0.928618285365 x2=0.980535228445 y2=0.939034952031
96) h x1=0.942803030303 y1=0.955883657136 x2=0.953219696970 y2=0.966192935486
97) h x1=0.912500000000 y1=0.986296446647 x2=0.922809278351 y2=0.996500528280
98) h x1=0.953672595520 y1=0.945130968963 x2=0.963876677153 y2=0.955231979064
99) h x1=0.927651515152 y1=0.971185641236 x2=0.937752525253 y2=0.981185641236
100) h x1=0.985176651306 y1=0.921829585372 x2=0.995176651306 y2=0.931730575471
101) h x1=0.927651515152 y1=0.981185641236 x2=0.937552505251 y2=0.990989562804
102) h x1=0.953672595520 y1=0.955231979064 x2=0.963476517089 y2=0.964940716929
103) h x1=0.937752525253 y1=0.971185641236 x2=0.947461263117 y2=0.980801025851
104) h x1=0.963876677153 y1=0.945130968963 x2=0.973492061768 y2=0.954654778487
105) h x1=0.985176651306 y1=0.931730575471 x2=0.994700460829 y2=0.941164537735
106) h x1=0.963876677153 y1=0.954654778487 x2=0.973310639417 y2=0.964000572880
107) h x1=0.937752525253 y1=0.980801025851 x2=0.947098319645 y2=0.990060285111
108) v x1=0.959824561404 y1=0.880316091954 x2=0.968998873330 y2=0.889575351213
109) v x1=0.840909090909 y1=0.840909090909 x2=0.850000000000 y2=0.850083402836
110) h x1=0.840909090909 y1=0.850083402836 x2=0.850000000000 y2=0.859092411845
111) h x1=0.840909090909 y1=0.859092411845 x2=0.849918099918 y2=0.868020983273
112) h x1=0.840909090909 y1=0.868020983273 x2=0.849837662338 y2=0.876870540795
113) h x1=0.840909090909 y1=0.876870540795 x2=0.849758648431 y2=0.885642470620
114) h x1=0.840909090909 y1=0.885642470620 x2=0.849681020734 y2=0.894338122794
115) h x1=0.840909090909 y1=0.894338122794 x2=0.849604743083 y2=0.902958812449
116) h x1=0.840909090909 y1=0.902958812449 x2=0.849529780564 y2=0.911505820996
117) h x1=0.840909090909 y1=0.911505820996 x2=0.849456099456 y2=0.919980397267
118) h x1=0.953672595520 y1=0.964940716929 x2=0.962147171792 y2=0.973344078273
119) v x1=0.491666666667 y1=0.833333333333 x2=0.500000000000 y2=0.841736694678
120) h x1=0.491666666667 y1=0.841736694678 x2=0.500000000000 y2=0.850001157488
121) h x1=0.491666666667 y1=0.850001157488 x2=0.499931129477 y2=0.858197878799
122) h x1=0.491666666667 y1=0.858197878799 x2=0.499863387978 y2=0.866327960100
123) h x1=0.491666666667 y1=0.866327960100 x2=0.499796747967 y2=0.874392476229
124) h x1=0.491666666667 y1=0.874392476229 x2=0.499731182796 y2=0.882392476229
125) h x1=0.491666666667 y1=0.882392476229 x2=0.499666666667 y2=0.890328984166
126) h x1=0.491666666667 y1=0.890328984166 x2=0.499603174603 y2=0.898202999914
127) h x1=0.491666666667 y1=0.898202999914 x2=0.499540682415 y2=0.906015499914
128) h x1=0.491666666667 y1=0.906015499914 x2=0.499479166667 y2=0.913767437898
129) h x1=0.491666666667 y1=0.913767437898 x2=0.499418604651 y2=0.921459745590
130) h x1=0.491666666667 y1=0.921459745590 x2=0.499358974359 y2=0.929093333377
131) h x1=0.491666666667 y1=0.929093333377 x2=0.499300254453 y2=0.936669090952
132) h x1=0.491666666667 y1=0.936669090952 x2=0.499242424242 y2=0.944187887945
133) h x1=0.491666666667 y1=0.944187887945 x2=0.499185463659 y2=0.951650574512
134) h x1=0.491666666667 y1=0.951650574512 x2=0.499129353234 y2=0.959057981920
135) h x1=0.491666666667 y1=0.959057981920 x2=0.499074074074 y2=0.966410923096
136) h x1=0.491666666667 y1=0.966410923096 x2=0.499019607843 y2=0.973710193169
137) h x1=0.724483639930 y1=0.985714285714 x2=0.731782910003 y2=0.992960662526
138) h x1=0.921428571429 y1=0.885221771790 x2=0.928674948240 y2=0.892416016395
139) h x1=0.826923076923 y1=0.992703536038 x2=0.834117321527 y2=0.999846393181
140) h x1=0.834117321527 y1=0.992703536038 x2=0.841260178670 y2=0.999795734620
141) h x1=0.841260178670 y1=0.992703536038 x2=0.848352377252 y2=0.999745789559
142) h x1=0.724483639930 y1=0.992960662526 x2=0.731525893451 y2=0.999953669519
143) h x1=0.973492061768 y1=0.945130968963 x2=0.980485068761 y2=0.952075413408
144) h x1=0.927651515152 y1=0.990989562804 x2=0.934595959596 y2=0.997886114529
145) h x1=0.973492061768 y1=0.952075413408 x2=0.980388613493 y2=0.958924728476
146) h x1=0.934595959596 y1=0.990989562804 x2=0.941445274664 y2=0.997792283893
147) h x1=0.980485068761 y1=0.945130968963 x2=0.987287789850 y2=0.951887725720
148) h x1=0.953672595520 y1=0.973344078273 x2=0.960429352277 y2=0.980055487669
149) h x1=0.962147171792 y1=0.964940716929 x2=0.968858581188 y2=0.971607383595
150) h x1=0.947098319645 y1=0.980801025851 x2=0.953764986312 y2=0.987423542408
151) h x1=0.980485068761 y1=0.951887725720 x2=0.987107585318 y2=0.958466673089
152) h x1=0.973492061768 y1=0.958924728476 x2=0.980071009137 y2=0.965460676189
153) h x1=0.650250626566 y1=0.993478260870 x2=0.656786574279 y2=0.999971767363
154) h x1=0.656786574279 y1=0.993478260870 x2=0.663280080772 y2=0.999929873773
155) h x1=0.663280080772 y1=0.993478260870 x2=0.669731693676 y2=0.999888517280
156) h x1=0.669731693676 y1=0.993478260870 x2=0.676141950086 y2=0.999847687621
157) h x1=0.676141950086 y1=0.993478260870 x2=0.682511376837 y2=0.999807374794
158) h x1=0.682511376837 y1=0.993478260870 x2=0.688840490761 y2=0.999767569046
159) h x1=0.688840490761 y1=0.993478260870 x2=0.695129798938 y2=0.999728260870
160) h x1=0.843589743590 y1=0.976310093415 x2=0.849839743590 y2=0.982521273540
161) h x1=0.843589743590 y1=0.982521273540 x2=0.849800923714 y2=0.988694113046
162) h x1=0.947461263117 y1=0.971185641236 x2=0.953634102623 y2=0.977320610561
163) v x1=0.843872229465 y1=0.959643426749 x2=0.849969790441 y2=0.965778396074
164) h x1=0.843872229465 y1=0.965778396074 x2=0.849969790441 y2=0.971839002134
165) h x1=0.970008912656 y1=0.939034952031 x2=0.976069518717 y2=0.945059048417
166) h x1=0.976069518717 y1=0.939034952031 x2=0.982093615102 y2=0.945022975983
167) h x1=0.987287789850 y1=0.945130968963 x2=0.993275813802 y2=0.951083349916
168) h x1=0.941445274664 y1=0.990989562804 x2=0.947397655617 y2=0.996906722568
169) h x1=0.962147171792 y1=0.971607383595 x2=0.968064331555 y2=0.977489736536
170) h x1=0.953764986312 y1=0.980801025851 x2=0.959647339253 y2=0.986648979068
171) h x1=0.987287789850 y1=0.951083349916 x2=0.993135743066 y2=0.956897303404
172) h x1=0.947397655617 y1=0.990989562804 x2=0.953211609105 y2=0.996769909625
173) h x1=0.993275813802 y1=0.945130968963 x2=0.999056160623 y2=0.950878095400
174) h x1=0.980485068761 y1=0.958466673089 x2=0.986232195198 y2=0.964180958803
175) h x1=0.973492061768 y1=0.965460676189 x2=0.979206347483 y2=0.971142494371
176) h x1=0.962147171792 y1=0.977489736536 x2=0.967828989973 y2=0.983139454051
177) h x1=0.478368836289 y1=0.994372294372 x2=0.484018553803 y2=0.999990271900
178) h x1=0.484018553803 y1=0.994372294372 x2=0.489636531332 y2=0.999958886551
179) h x1=0.489636531332 y1=0.994372294372 x2=0.495223123510 y2=0.999927849928
180) h x1=0.911794871795 y1=0.744444444444 x2=0.917350427350 y2=0.749969306323
181) h x1=0.917350427350 y1=0.744444444444 x2=0.922875289229 y2=0.749938949939
182) h x1=0.922875289229 y1=0.744444444444 x2=0.928369794723 y2=0.749908925319
183) h x1=0.928369794723 y1=0.744444444444 x2=0.933834275598 y2=0.749879227053
184) h x1=0.933834275598 y1=0.744444444444 x2=0.939269058206 y2=0.749849849850
185) h x1=0.968064331555 y1=0.971607383595 x2=0.973469736960 y2=0.976983727681
186) h x1=0.953764986312 y1=0.986648979068 x2=0.959141330398 y2=0.991996572651
187) h x1=0.993275813802 y1=0.950878095400 x2=0.998623407385 y2=0.956197244336
188) h x1=0.459500911761 y1=0.994708994709 x2=0.464820060697 y2=1.000000000000
189) h x1=0.464820060697 y1=0.994708994709 x2=0.470111065988 y2=0.999972152604
190) h x1=0.470111065988 y1=0.994708994709 x2=0.475374223883 y2=0.999944596803
191) v x1=0.844780219780 y1=0.942099567100 x2=0.849988553114 y2=0.947335169194
192) h x1=0.844780219780 y1=0.947335169194 x2=0.849988553114 y2=0.952516516344
193) v x1=0.939269058206 y1=0.744444444444 x2=0.944423697382 y2=0.749625791595
194) h x1=0.844780219780 y1=0.952516516344 x2=0.849934858955 y2=0.957644721472
195) h x1=0.994700460829 y1=0.931730575471 x2=0.999828665958 y2=0.936832616287
196) h x1=0.994700460829 y1=0.936832616287 x2=0.999802501646 y2=0.941908758419
197) h x1=0.987287789850 y1=0.956897303404 x2=0.992363931982 y2=0.961947808455
198) h x1=0.973492061768 y1=0.971142494371 x2=0.978542566819 y2=0.976167619999
199) h x1=0.980485068761 y1=0.964180958803 x2=0.985510194390 y2=0.969180958803
200) h x1=0.942803030303 y1=0.966192935486 x2=0.947803030303 y2=0.971168059864

Other modifications:

And packing with horizontal rectangles only: