Water retention on mathematical surfaces

Not to be confused with Wetting § Ideal solid 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

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.

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References

1 2 3 4 5 Craig Knecht, http://www.knechtmagicsquare.paulscomputing.com
1 2 Al Zimmermann http://www.azspcs.net/Contest/MagicWater/FinalReport
1 2 3 Harvey Heinz, http://www.magic-squares.net/square-update-2.htm#Knecht
1 2 3 Harry White, http://budshaw.ca/Download.html
↑ Walter Trump http://www.trump.de/magic-squares/
↑ Johan Ofverstedt,http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-176018
↑ Harry White, http://budshaw.ca/Most-perfect.html
↑ 2x2 planar subsets,https://oeis.org/A270205
↑ Harvey Heinz,http://www.magic-squares.net/square-update.htm
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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)
↑ 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)
↑ 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)
1 2 3 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.
↑ 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.
1 2 Schrenk, K. J.; N. A. M Araújo; R. M. Ziff; H. J. Herrmann (2014). "Retention capacity of correlated surfaces". arXiv:1403.2082.
↑ 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.

External links

Hugo Pfoertner. "Maximum water retention of a magic square of order n", A201126 with links to magic square pictures
Hugo Pfoertner. "Maximum water retention of a semi-magic square of order n", A201127
Discussion site for the Zimmermann problems
Greg Ross. Futility Closet
Maximum retention for associative magic squares
Polyominoe enumeration and lake patterns
Upgrade the model from 2D to 3D

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