Water retention on mathematical surfaces

Illustration of water retention on a surface.

Water retention on mathematical surfaces refers to the water caught in ponds on a surface of cells of various heights on a regular array such as a square lattice, where water is rained down on every cell in the system. The boundaries of the system are open and allow water to flow out. Water will be trapped in ponds, and eventually all ponds will fill to their maximum height, with any additional water flowing over spillways and out the boundaries of the system. The problem is to find the amount of water trapped or retained for a given surface. This has been studied extensively for two mathematical surfaces: magic squares and random surfaces.

Magic squares

A 5x5 magic square with the maximal retention.

Magic squares have been studied for over 2000 years. In 2007, the idea of studying the water retention on a magic square was proposed.^[1] In 2010, Al Zimmermann's programing contest^[2] produced the presently known maximum retention values for magic squares order 4 to 28.^[3] Computing tools used to investigate and illustrate this problem are found here.^[4][5][6]

There are 4,211,744 different retention patterns for the 7x7 square. A combination of a lake and ponds is best for attaining maximum retention. No known patterns for maximum retention have an island in a pond or lake.^[1]

136 units of retention 1 34 36 32 31 39 2 25 20 5 15 14 49 47 24 12 45 28 16 13 37 38 9 27 40 29 11 21 41 8 17 23 46 18 22 43 48 10 7 6 19 42 3 44 35 30 33 26 4 island spillway

365 units of retention 5 33 44 39 34 18 2 20 49 8 15 11 48 24 37 14 23 29 30 6 36 43 12 22 17 26 10 45 38 1 27 21 32 16 40 28 47 9 13 7 46 25 4 19 42 41 35 31 3 lake

405 units of retention 4 25 46 19 31 32 18 27 44 16 45 12 2 29 47 10 14 6 48 20 30 21 43 1 42 5 41 22 33 11 49 3 24 15 40 34 7 13 37 17 39 28 9 35 36 23 38 26 8 pond

417 units of retention 13 30 22 41 39 20 10 31 15 48 7 4 45 25 27 44 17 18 21 8 40 23 42 5 29 9 43 24 34 1 47 6 49 2 36 35 11 3 46 16 38 26 12 32 33 28 37 19 14 best for max retention

Maximum-retention magic squares for orders 7-9 are shown below:^[3]

418 units of retention 2 22 38 40 39 19 15 29 45 13 12 4 44 28 42 17 20 21 26 8 41 25 46 10 18 6 43 27 32 3 49 7 48 1 35 31 9 11 47 16 37 24 14 33 34 30 36 23 5 Hermann Jurksch April 6, 2010

797 units of retention 1 26 41 39 50 47 40 16 36 43 21 61 9 7 51 32 42 6 62 15 57 25 5 48 29 58 14 23 27 56 4 49 52 13 20 33 22 18 64 38 53 17 35 24 8 60 19 44 37 63 12 11 59 2 46 30 10 34 55 54 28 45 31 3 Hermann Jurksch April 5, 2010

1,408 units of retention 1 37 57 47 66 50 53 55 3 43 59 11 77 25 75 10 15 54 58 7 78 21 34 22 81 12 56 45 73 14 23 32 30 27 74 51 69 17 39 28 52 36 35 26 67 44 72 18 38 42 13 20 76 46 64 6 80 16 31 24 79 8 61 40 65 9 70 19 71 4 62 29 5 33 63 49 68 48 60 41 2 Walter Trump June 12, 2010

The figures below show the 10x10 magic square. Is it possible to look at the patterns above and predict what the pattern for maximum retention for the 10x10 square will be? No theory has been developed that can predict the correct combination of lake and ponds for all orders, however some principles do apply. The first color-coded figure shows a design principle of how the largest available numbers are placed around the lake and ponds. The second and third figures show promising patterns that were tried but did not achieve maximum retention.^[1]

2267 units of retention 1 41 79 58 75 63 64 68 52 4 47 80 5 81 8 90 87 6 76 25 78 7 35 23 97 44 36 100 13 72 60 82 19 91 39 29 27 14 93 51 73 21 98 32 40 48 33 59 16 85 65 89 46 26 50 62 34 30 20 83 56 88 43 28 31 37 22 42 92 66 70 11 99 17 55 38 45 94 9 67 53 71 12 95 24 10 96 18 77 49' 2 15 69 54 86 84 61 74 57 3 color key: > > > >

2251 units of retention 1 50 66 46 86 54 77 74 48 3 53 68 17 97 32 91 25 4 78 40 67 13 98 28 35 24 96 52 16 76 58 93 15 41 36 21 51 92 23 75 85 30 29 34 45 64 37 26 99 56 55 95 10 31 63 44 22 89 14 82 71 20 90 60 33 27 100 7 18 79 70 6 59 88 19 87 8 38 83 47 43 73 72 11 94 12 9 84 65 42 2 57 49 69 62 81 80 39 61 5

2228 units of retention 1 45 79 53 87 83 55 72 27 3 51 80 20 95 11 40 89 14 74 31 78 19 90 42 15 56 33 94 5 73 48 97 30 9 64 26 36 43 91 41 85 28 54 22 68 57 61 23 25 82 86 13 39 62 21 52 46 60 38 88 66 99 10 49 59 32 12 24 96 58 70 9 98 6 34 67 17 92 37 75 18 71 16 100 65 8 93 7 77 50 2 44 69 47 81 84 63 76 35 4

Several orders have more than one pattern for maximum retention. The figure below shows the two patterns for the 11x11 magic square with the apparent maximum retention of 3,492 units:^[3]

Pattern #1 1 55 81 69 103 47 83 64 99 67 2 59 89 13 106 17 111 18 102 8 100 48 82 14 119 36 45 53 117 19 76 12 98 73 107 31 43 46 23 37 112 30 101 68 104 20 56 61 54 60 52 38 121 21 84 105 27 25 50 80 70 66 16 49 108 75 74 109 35 39 78 58 41 22 120 5 90 86 10 115 57 34 28 40 114 26 96 65 77 91 33 116 29 32 118 24 42 15 94 6 87 92 9 113 110 11 97 7 95 44 4 62 71 85 72 79 88 63 93 51 3

Pattern #2 1 36 97 69 86 74 64 89 90 62 3 68 99 10 101 8 108 109 12 11 92 53 98 5 31 15 118 28 58 116 61 91 50 77 100 21 113 22 29 35 66 117 9 82 87 6 121 32 51 41 72 39 38 114 70 79 105 46 33 52 71 60 47 49 25 104 75 106 54 23 81 55 56 45 48 26 102 88 7 120 67 42 40 63 43 34 110 57 76 96 59 107 27 30 37 24 119 13 83 18 95 19 17 111 115 14 112 20 85 65 4 16 93 94 73 80 103 78 84 44 2

The most-perfect magic squares require all (n-1)^2 or in this case all 121 2x2 planar subsets to have the same sum. ( a few examples flagged with yellow background, red font). Increased internal complexity reduces retention.^[7][8]

Before 2010 if you wanted an example of a magic square larger than 5x5 you had to follow clever construction rules that provided very isolated examples. The 13x13 pandiagonal magic square below is such an example. Harry White's CompleteSquare Utility ^[4] allows anyone to use the magic square as a potter would use a lump of clay. The second image shows a 14x14 magic square that was molded to form ponds that write the 1514 - 2014 dates. The animation notes how the surface was sculptured to fill all ponds to capacity before the water flows off the square. This square honors the 500th anniversary of Durer's famous magic square in Melencolia I.

The figure below is a 15x15 bordered magic square with zero water retention.^[4] This figure also provides an example of a square and its complement that have the same pattern of retention. There are 137 order 4 and 3,254,798 order 5 magic squares that do not retain water.^[1]

212 198 200 202 204 206 208 13 12 10 8 6 4 2 210 27 186 174 176 178 180 182 39 38 36 34 32 30 184 199 25 51 164 154 156 158 160 61 60 58 56 54 162 175 201 23 49 71 146 138 140 142 79 78 76 74 144 155 177 203 21 47 69 87 132 126 128 93 92 90 130 139 157 179 205 19 45 67 85 99 122 118 103 102 120 127 141 159 181 207 17 43 65 83 97 107 116 109 114 119 129 143 161 183 209 15 41 63 81 95 105 111 113 115 121 131 145 163 185 211 215 189 167 149 135 125 112 117 110 101 91 77 59 37 11 217 191 169 151 137 106 108 123 124 104 89 75 57 35 9 219 193 171 153 96 100 98 133 134 136 94 73 55 33 7 221 195 173 82 88 86 84 147 148 150 152 80 53 31 5 223 197 64 72 70 68 66 165 166 168 170 172 62 29 3 225 42 52 50 48 46 44 187 188 190 192 194 196 40 1 16 28 26 24 22 20 18 213 214 216 218 220 222 224 14

16 x 16 associative magic square retaining 17840 units. The lake in the first image looks a little uglier than common. Jarek Wroblewski notes that good patterns for maximum retention will have equal or near equal number of retaining cells on each peripheral edge ( in this case 7 cells on each edge) ^[2] The second image is doctored, shading in the top and bottom 37 values.

The figure below is a 17x17 Luo-Shu format magic square.^[9] The Luo-Shu format construction method seems to produce a maximum number of ponds. The drainage path for the cell in green is long eventually spilling off the square at the yellow spillway cell.

The figure to the right shows what information can be derived from looking at the actual water content for each cell. Only the 144 values are highlighted to keep the square from looking too busy. Focusing on the green cell with a base value 7, the highest obstruction on the path out is its neighbor cell with the value of 151 (151-7=144 units retained). Water rained into this cell exits the square at the yellow 10 cell.

137 282 121 266 105 250 89 234 73 218 57 202 41 186 25 170 9 10 138 283 122 267 106 251 90 235 74 219 58 203 42 187 26 154 155 11 139 284 123 268 107 252 91 236 75 220 59 204 43 171 27 28 156 12 140 285 124 269 108 253 92 237 76 221 60 188 44 172 173 29 157 13 141 286 125 270 109 254 93 238 77 205 61 189 45 46 174 30 158 14 142 287 126 271 110 255 94 222 78 206 62 190 191 47 175 31 159 15 143 288 127 272 111 239 95 223 79 207 63 64 192 48 176 32 160 16 144 289 128 256 112 240 96 224 80 208 209 65 193 49 177 33 161 17 145 273 129 257 113 241 97 225 81 82 210 66 194 50 178 34 162 1 146 274 130 258 114 242 98 226 227 83 211 67 195 51 179 18 163 2 147 275 131 259 115 243 99 100 228 84 212 68 196 35 180 19 164 3 148 276 132 260 116 244 245 101 229 85 213 52 197 36 181 20 165 4 149 277 133 261 117 118 246 102 230 69 214 53 198 37 182 21 166 5 150 278 134 262 263 119 247 86 231 70 215 54 199 38 183 22 167 6 151 279 135 136 264 103 248 87 232 71 216 55 200 39 184 23 168 7 152 280 281 120 265 104 249 88 233 72 217 56 201 40 185 24 169 8 153

00 144 144 144 144 144 144 128 127 144 144 144 144 144 128 127 144 144 144 144 128 127 127 127 144 144 144 128 127 127 127 144 144 128 127 127 127 127 127 144 128 127 127 127 127 127 128 127 127 127 127 127 127 127 127 127 127 127 127 127 144 127 127 127 127 127 127 144 144 127 127 127 127 144 144 144 127 127 127 127 144 144 144 144 127 127 144 144 144 144 144 127 127 144 144 144 144 144 144 144 144 144 144 144 144 144 00

The computer age now allows for the exploration of the physical properties of magic squares of any order. The figure below shows the largest magic square studied in the contest. For L > 20 the number of variables/ equations increases to the point where it makes the pattern for maximum retention predictable.

L = 28: 219822 units of retention

1	5	259	659	257	713	712	282	256	283	657	255	656	284	726	725	254	285	654	253	286	55	673	674	471	645	7	3
9	640	664	25	717	26	27	716	715	714	28	668	29	744	30	31	730	743	50	681	680	679	51	52	678	206	646	11
265	665	355	722	496	618	71	484	95	121	721	400	774	418	130	176	293	541	749	479	106	175	389	148	230	682	43	644
663	14	724	356	313	513	75	189	198	449	213	775	87	478	539	139	326	60	451	750	461	566	141	442	638	477	677	276
266	720	164	572	354	226	491	171	512	117	776	247	244	503	435	85	629	406	144	634	751	592	462	125	134	514	44	672
711	15	565	116	100	357	579	112	637	777	108	469	433	546	80	559	525	468	526	227	146	752	368	557	328	212	46	671
710	16	153	174	222	119	353	627	778	64	297	456	544	474	178	473	410	563	515	331	403	387	753	402	569	304	45	670
709	17	70	325	168	509	445	779	166	366	401	83	92	482	129	338	408	492	585	529	369	298	424	754	582	519	676	275
267	719	156	103	455	531	780	391	358	537	76	142	367	309	522	245	320	437	632	386	545	497	224	123	755	161	675	277
264	718	444	600	508	781	196	553	65	352	488	344	624	104	216	551	98	616	370	294	233	101	416	490	109	756	47	652
662	18	723	417	782	310	564	606	420	483	359	518	548	246	475	58	628	385	571	69	149	223	335	235	86	113	733	274
263	669	218	783	127	429	581	77	399	136	88	351	602	538	636	635	371	220	74	570	99	633	543	498	502	173	48	727
661	19	784	407	179	184	195	609	393	495	203	567	360	576	394	384	388	137	625	154	523	229	489	485	219	314	738	279
268	748	597	307	505	615	441	315	583	562	194	542	446	350	372	588	316	443	120	162	89	102	560	317	110	329	737	272
729	20	521	177	232	340	128	411	152	122	334	241	605	383	361	412	578	202	619	73	611	549	589	587	432	568	736	278
262	746	68	580	242	187	558	183	398	601	594	182	373	296	460	349	332	556	205	419	614	323	547	586	207	114	735	273
269	745	458	131	111	78	337	610	532	612	622	382	59	365	554	448	362	613	82	574	172	493	466	126	145	630	734	280
261	747	158	465	598	221	459	214	524	167	374	608	533	409	319	330	595	348	181	428	305	453	584	199	61	765	33	651
660	21	773	536	561	94	345	165	204	381	621	528	447	211	500	135	452	342	363	301	396	527	185	225	764	306	666	281
270	694	517	772	392	431	312	240	375	190	617	151	91	324	333	520	231	215	511	347	540	238	97	763	413	707	49	650
260	693	105	405	771	550	295	380	302	336	311	620	234	133	427	197	516	150	90	607	364	425	762	486	67	530	703	271
53	692	300	163	631	770	376	191	157	552	414	415	555	422	626	590	339	507	79	188	147	761	430	308	436	132	702	54
683	22	397	423	535	379	769	155	421	494	322	454	390	217	510	623	107	200	591	186	760	341	346	593	237	115	24	696
684	23	228	118	377	575	303	768	327	534	487	573	438	472	457	599	464	439	143	759	604	138	160	72	395	124	32	697
480	691	209	378	440	504	140	501	767	81	201	159	404	210	467	577	57	169	758	193	426	470	93	596	639	180	701	499
648	258	695	299	192	208	481	321	318	766	463	96	63	506	84	236	239	757	343	708	450	243	170	434	603	706	62	641
10	649	38	690	39	37	689	688	687	40	732	36	742	741	740	739	731	41	667	35	705	42	34	13	704	66	643	12
2	6	647	287	686	685	288	252	251	658	289	728	250	249	290	248	291	655	292	653	56	698	699	700	476	642	8	4
Jarek Wroblewski March 24, 2010

2014 brought the ability to write a unlimited amount of text on the mathematical surface of a magic square.^[1][4]

Random surfaces

Water retention on a random surface of 10 levels.

Another system in which the retention question has been studied is a surface of random heights. Here one can map the random surface to site percolation, and each cell is mapped to a site on the underlying graph or lattice that represents the system. Using percolation theory, one can explain many properties of this system. It is an example of the invasion percolation model in which fluid is introduced in the system from any random site.^[10][11][12]

In hydrology, one is concerned with runoff and formation of catchments.^[13] The boundary between different drainage basin (watersheds in North America) forms a drainage divide with a fractal dimension of about 1.22.^{[14] [15] [16]}

The retention problem can be mapped to standard percolation.^[17][18][19] For a system of five equally probable levels, for example, the amount of water stored R₅ is just the sum of the water stored in two-level systems R₂(p) with varying fractions of levels p in the lowest state:

R₅ = R₂(1/5) + R₂(2/5) + R₂(3/5) + R₂(4/5)

Typical two-level systems 1,2 with p = 0.2, 0.4, 0.6, 0.8 are shown on the right (blue: wet, green: dry, yellow: spillways bordering wet sites). The net retention of a five-level system is the sum of all these. The top level traps no water because it is far above the percolation threshold for a square lattice, 0.592746.

The retention of a two-level system R₂(p) is the amount of water connected to ponds that do not touch the boundary of the system. When p is above the critical percolation threshold p _c, there will be a percolating cluster or pond that visits the entire system. The probability that a point belongs to the percolating or "infinite" cluster is written as P_∞ in percolation theory, and it is related to R₂(p) by R₂(p)/L² = p − P_∞ where L is the size of the square. Thus, the retention of a multilevel system can be related to a well-known quantity in percolation theory.

To measure the retention, one can use a flooding algorithm in which water is introduced from the boundaries and floods through the lowest spillway as the level is raised. The retention is just the difference in the water level that a site was flooded minus the height of the terrain below it.

Besides the systems of discrete levels described above, one can make the terrain variable a continuous variable say from 0 to 1. Likewise, one can make the surface height itself be a continuous function of the spatial variables. In all cases, the basic concept of the mapping to an appropriate percolation system remains.

A curious result is that a square system of n discrete levels can retain more water than a system of n+1 levels, for sufficiently large order L > L*. This behavior can be understood through percolation theory, which can also be used to estimate L* ≈ (p - p_c)^-ν where ν = 4/3, p = i*/n where i* is the largest value of i such that i/n < p_c, and p_c = 0.592746 is the site percolation threshold for a square lattice. Numerical simulations give the following values of L*, which are extrapolated to non-integer values. For example, R₂ < R₃ for L ≤ 51, but R₂ > R₃ for L ≥ 52:^[17]

n	n+1	L*	Retention at L*
2	3	51.12	790
4	5	198.1	26000
7	8	440.3	246300
9	10	559.1	502000
12	13	1390.6	428850
14	15	1016.3	2607000

As n gets larger, crossing become less and less frequent, and the value of L* where crossing occurs is no longer a monotonic function of n.

The retention when the surface is not entirely random but correlated with a Hurst exponent H is discussed in.^[19]

Algorithms

The following time line shows the application of different algorithms that have expanded the size of the square that can be evaluated for retention

2007 Define all neighbor-avoiding walks from each interior cell to the exterior and then sort all those paths for the least obstruction or cell value. The least obstruction value minus the interior cell value provides the water retention for that interior cell (negative values are set to a retention value of 0). The number of neighbor-avoiding walks to be evaluated grows exponentially with the square size and thus limits this methodology to L < 6.^[1]

2009 Flooding algorithm - water is introduced from the boundaries and floods through the lowest spillway as the level is raised. The retention is just the difference in the water level that a site was flooded minus the height of the terrain below it. The flooding algorithm allows for the evaluation of water retention up to L < 10,000.^[17] This algorithm is similar to Meyer's flooding algorithm that has been used in analysis of topographical surfaces.

2011 With the realization that an n-level system can be broken down into a collection of two-level systems with varying probabilities, standard percolation algorithms can be used to find the retention as simply the total number of sites at the lower level minus the draining regions (clusters of low-level sites touching the boundary). A novel application of the Hoshen-Kopelman algorithm in which both rows and columns are added one at a time allows L to be very large (up to 10⁹), but computing time considerations limits L to the order of 10⁷.^[20]

Paths that drain water off the square, used in the neighbor-avoiding walk algorithm

The panel below from left to right shows: 1) the three unique interior positions for the 5x5 square; 2 & 4) correct paths off the square in grey for the interior corner cell in red; 3) incorrect path in grey as the water cannot travel on the diagonals; 5) this path is correct but there is a short-circuit possible between the grey cells. Neighbor-avoiding walks define the unique or non-redundant paths that drain water off the square.

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 1

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

See also

• Drainage divide

References

1. Craig Knecht, http://www.knechtmagicsquare.paulscomputing.com
2. Al Zimmermann http://www.azspcs.net/Contest/MagicWater/FinalReport
3. Harvey Heinz, http://www.magic-squares.net/square-update-2.htm#Knecht
4. Harry White, http://users.eastlink.ca/~sharrywhite/Download.html
5. Walter Trump http://www.trump.de/magic-squares/
6. Johan Ofverstedt,http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-176018
7. Harry White, http://users.eastlink.ca/~sharrywhite/Most-perfect.html
8. 2x2 planar subsets,https://oeis.org/A270205
9. Harvey Heinz,http://www.magic-squares.net/square-update.htm
10. Chayes, J. T.; L. Chayes; C. M. Newman (1985). "The stochastic geometry of invasion percolation". Communications in Mathematical Physics 101 (3): 383–407. Bibcode:1985CMaPh.101..383C. doi:10.1007/BF01216096.
11. Damron, Michael; Artëm Sapozhnikov; Bálint Vágvölgyi (2009). "Relations between invasion percolation and critical percolation in two dimensions". Annals of Probability 37 (6): 2297–2331. doi:10.1214/09-AOP462.
12. van den Berg, Jacob; Antal Járai; Bálint Vágvölgyi (2007). "The size of a pond in 2D invasion percolation". Electron. Comm. In Probab. 12: 411–420. doi:10.1214/ECP.v12-1327.
13. Tetzlaff, D.; McDonnell, J. J.; Uhlenbrook, S.; McGuire, K. J.; Bogaart, P. W.; Naef, F.; Baird, A. J.; Dunn, S. M.; Soulsby, C. (2011). "Conceptualizing catchment processes: simply too complex?". Hydrological Processes 22 (11): 1099–1085. Bibcode:2008HyPr...22.1727T. doi:10.1002/hyp.7069.
14. Fehr, E.; D. Kadau, N. A. M. Araújo, J. S. Andrade Jr., and H. J. Herrmann (2011). "Scaling Relations for Watersheds". Physical Review E 84 (3): 036116. arXiv:1106.6200. Bibcode:2011PhRvE..84c6116F. doi:10.1103/PhysRevE.84.036116. Cite uses deprecated parameter |coauthors= (help)
15. Schrenk, K. J.; N. A. M. Araújo, J. S. Andrade Jr., and H. J. Herrmann (2012). "Fracturing Ranked Surfaces". Scientific Reports 2: 348. arXiv:1103.3256. Bibcode:2012NatSR...2E.348S. doi:10.1038/srep00348. Cite uses deprecated parameter |coauthors= (help)
16. Fehr, E.; D. Kadau, J. S. Andrade Jr., and H. J. Herrmann (2011). "Impact of Perturbations on Watersheds". Physical Review Letters 106 (4): 048501. arXiv:1101.5890. Bibcode:2011PhRvL.106d8501F. doi:10.1103/PhysRevLett.106.048501. Cite uses deprecated parameter |coauthors= (help)
17. Knecht, Craig; Walter Trump; Daniel ben-Avraham; Robert M. Ziff (2012). "Retention capacity of random surfaces". Physical Review Letters 108 (4): 045703. arXiv:1110.6166. Bibcode:2012PhRvL.108d5703K. doi:10.1103/PhysRevLett.108.045703.
18. Baek, Seung Ki; Beom Jun Kim (2012). "Critical Condition of the Water-Retention Model". Physical Review E 85: 032103. arXiv:1111.0425. Bibcode:2012PhRvE..85c2103B. doi:10.1103/PhysRevE.85.032103.
19. Schrenk, K. J.; N. A. M Araújo; R. M. Ziff; H. J. Herrmann (2014). "Retention capacity of correlated surfaces". arXiv:1403.2082.
20. Hoshen, Joseph (1998). "On the application of the enhanced Hoshen-Kopelman algorithm for image analysis". Pattern Recognition Letters 19 (7). doi:10.1016/S0167-8655(98)00018-x.

Further reading

• Pickover, Clifford (2002). The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press. ISBN 0-691-11597-4.
• Stauffer, Dietrich; Aharony, A. (1994). Introduction to Percolation Theory. London Bristol, PA: Taylor & Francis. ISBN 978-0-7484-0253-3.

External links

• Pfoertner, Hugo. "Maximum water retention of a magic square of order n". ,  A201126 with links to magic square pictures
• Pfoertner, Hugo. "Maximum water retention of a semi-magic square of order n". ,  A201127
• Discussion site for the Zimmermann problems
• Ross, Greg. "Futility Closet". , http://www.futilitycloset.com/2013/03/30/stormy-weather-3/
• Maximum retention for associative magic squares
• Polyominoe enumeration and lake patterns