|
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 == Magic squares have been studied for over 2000 years. In 2007, the idea of studying the water retention on a magic square was proposed.〔Craig Knecht, http://www.knechtmagicsquare.paulscomputing.com〕 In 2010, Al Zimmermann's programing contest〔Al Zimmermann http://www.azspcs.net/Contest/MagicWater/FinalReport〕 produced the presently known maximum retention values for magic squares order 4 to 28.〔Harvey Heinz, http://www.magic-squares.net/square-update-2.htm#Knecht〕 Computing tools used to investigate and illustrate this problem are found here.〔Harry White, http://users.eastlink.ca/~sharrywhite/Download.html〕〔Walter Trump http://www.trump.de/magic-squares/〕〔Johan Ofverstedt,http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-176018〕 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.〔 Maximum-retention magic squares for orders 7-9 are shown below:〔 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.〔 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:〔 The most-perfect magic squares require all 2x2 cell blocks to have the same sum. ( a few examples flagged with yellow background, red font). Increased internal complexity reduces retention.〔Harry White, http://users.eastlink.ca/~sharrywhite/Most-perfect.html〕 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 〔 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.〔 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.〔 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) 〔 The second image is doctored, shading in the top and bottom 37 values. The figure below is a 17x17 Luo-Shu format magic square.〔Harvey Heinz,http://www.magic-squares.net/square-update.htm〕 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. 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. 2014 brought the ability to write a unlimited amount of text on the mathematical surface of a magic square.〔〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Water retention on mathematical surfaces」の詳細全文を読む スポンサード リンク
|