Wikipedia:Reference desk/Archives/Mathematics/2015 September 10

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September 10

Characteristics of a slice through a cube.

A slice through a Cube can be one of four polygons (triangle, quadrilateral, pentagon, hexagon). What I'm trying to figure out is what characteristics would a specific one of those polygons would have to have in order to be able to say specifically that it can or can not be a slice. For example, I believe that a triangular slice would have to be either Acute or Right, and no convex polygons could be made. I don't think a regular Pentagon slice could exist, but I'm really not sure what inequality could be generated that would tell me that. Similarly, I don't think a hexagon with angles of 175, 107, 107, 107, 107, 107 could be made either, but again,I don't know what inequality characteristic could say that. Note, the lengths of the sides also affect this, while a regular hexagon could be made, one with a large amount of length added to opposite sides could not be even though all the angels were the same. Ideas?Naraht (talk) 04:58, 10 September 2015 (UTC)

The triangle can't be right, so it must be acute. No idea what general limitations apply to more-gon sections. --CiaPan (talk) 06:43, 10 September 2015 (UTC)

It's easy to see that such a quadrilateral must be a trapezoid, such a pentagon must have two pairs of parallel sides and such a hexagon must have three pairs of parallel sides, and this remains the case for slices of a general parallelepiped. That at least eliminates the regular pentagon and the hexagon given above. The parallel sides criteria and convexity may be sufficient criteria for the polygonal slices of a parallelepiped. Any slice of a convex polyhedron is convex so that eliminates concave hexagons with parallel sides. --RDBury (talk) 08:55, 10 September 2015 (UTC)

"no convex polygons could be made". Oh yes. The cube is convex and the plane is convex and the intersection between two convex sets is convex, so all the polygons that can be made by slicing a cube by a plane are convex. Bo Jacoby (talk) 19:20, 11 September 2015 (UTC).

Choice of Parameters for Linear Congruential Generator

For a Linear_congruential_generator with a given modulo m and non-zero increment c, will all values of a that satisfy the conditions of the Hull-Dobell theorem (as it is stated in the article) result in a full period? If so, is there any reason to choose one value of a over another? For example, if m = 2²⁰ and a = 3², then the Hull-Dobell conditions are met, since (1) m and a are relatively prime, (2) a - 1 is divisible by 2, and (3) both a - 1 and m are divisible by 4. I'd like to use these values, because my program has very tight memory constraints, and it will keep the product computed before the modulo step smaller; however, I don't know if another choice of a would have better properties (or what that would mean, given that a = 3² gives a full period). OldTimeNESter (talk) 20:41, 10 September 2015 (UTC)

I think that all values that satisfy the conditions of the Hull-Dobell theorem will have a full period, because that seems to be exactly the theorem. But note that the first condition is that m and c are relatively prime, not m and a.

Some choices are definitely better than others: for example, a = c = 1 is a bad choice despite satisfying the conditions of the theorem. A good LCG should do well on the spectral test, but I don't know how to find values with that property in practice. I suspect that small multipliers in general are not very good, because they would be recommended more often if they were. -- BenRG (talk) 03:41, 11 September 2015 (UTC)

Pentagonal tiling

Hi, I have recently been looking at the very interesting article on pentagonal tiling, but somehow I cannot seem to see which of the fifteen types the very simple pattern here belongs to. Could someone show me which one it is? 109.153.229.252 (talk) 23:29, 10 September 2015 (UTC)

Type 1. Tiling types are numbered based on the pentagons they contain, and not the specific arrangement, In some cases, such as the example shown, the pentagons can be arranged in multiple ways. We don't enumerate all the specific ways, but just the specific classes of pentagons. Your pentagons in the example given can also be made to fit a Type 1 arrangement. A new tiling type is only declared when it contains pentagons not belonging to any other known type. Dragons flight (talk) 00:04, 11 September 2015 (UTC)

• And indeed it is shown in our article, at "cmm (2*22)". —Tamfang (talk) 03:30, 11 September 2015 (UTC)

So a new and "topologically different" (if that is the correct term) arrangement of a pentagon that happens to fit one of the 15 profiles doesn't count as a new tiling? That is not a very helpful or sensible way of classifying them, in my view. I look at things like number of lines meeting at each vertex, and the spatial way these join together -- things that are integral to the pattern and cannot be changed just by distorting pentagons. For example, my pattern has nodes with four vertices, and also has T-junctions, and most of the illustrated type-1 patterns dp not have that combination. That makes my tiling different, in my view. However, now that I zoom in, it does appear that my tiling may be a distorted version of the fourth type-1 example on the far right. 109.153.229.252 (talk) 00:48, 11 September 2015 (UTC) — Preceding unsigned comment added by 109.153.229.252 (talk)

It isn't that there are 15 types of tiling, it is, as stated in the article "Fifteen types of convex pentagons are known to tile the plane monohedrally". The number of topologically distinct ways in which those types can be tiled is larger. The fact that the new one is a 15th is because a pentagon with those angles and side lengths didn't fit into any of the criteria of the other 14. And please do ask on the talk page for the article as well. Note, at this point while I'm fascinated by the choices presented, some of this is rapidly approaching WP:OR. Naraht (talk) 06:14, 11 September 2015 (UTC)

Because some of the tiles permit partial rotations of local groups, the number of distinct ways to tile the plane with pentagons is literally infinite. Hence it makes more sense to enumerate classes of pentagons rather than classes of arrangements. Dragons flight (talk) 06:58, 11 September 2015 (UTC)

As an example, with type 3, a specific generated (from 3 Pentagons) Hexagon may be rotated 180 as well as all making up a hex grid with separation by two squares, or with a separation of 3 sequares, or ...Naraht (talk) 07:30, 11 September 2015 (UTC)

• Thanks for the replies. I still feel that a "fundamentally new" method of tiling, using some suitable understanding of "fundamentally new" to exclude "obvious" extensions of known patterns, ought to count as a new discovery even if it happens to make use of a pentagon type that is already known to tile in a different way. For example, what if the exotic type 15 had just happened to need a pentagon which had an existing different known tiling? Would it not have been classed as a new type of something? 109.157.10.150 (talk) 19:40, 11 September 2015 (UTC)

The specific pattern you linked to is in the article (as mentioned above) and to be clear it's this file here... and they happen to look like monopoly houses. -8billionthaveragejoe (talk) 20:41, 11 September 2015 (UTC)

• (OP) ... Oh, just one more please. Which type is this? I do seem to find it very hard to match them up. 109.157.10.150 (talk) 20:48, 11 September 2015 (UTC)

Also Type 1. Dragons flight (talk) 21:00, 11 September 2015 (UTC)

But hold on ... does "Type 1" include any pentagon with two parallel sides? 109.157.10.150 (talk) 21:54, 11 September 2015 (UTC)

Yes. Dragons flight (talk) 21:59, 11 September 2015 (UTC)

Thanks. So any pattern that I create that uses a pentagon with two parallel sides is "Type 1", regardless of how they fit together? I'm sorry, but I think the rules of this game are silly. The most important thing which defines a tiling is how the tiles fit together (allowing for the fact, as mentioned above, that there may be families of patterns created by distortion, rotation, etc.). The whole notion of classifying tiling patterns purely by the type of pentagon, without any account taken of how the pentagons are fitted together, makes very little sense to me. 109.157.10.150 (talk) 22:07, 11 September 2015 (UTC)

Individual arrangements can be labelled by their wallpaper group (e.g. c2mm (2*22)). Many examples of such labelling appear in pentagonal tiling, and as mentioned above, the possibilities are literally infinite which makes enumerating them less interesting to researchers. Dragons flight (talk) 22:14, 11 September 2015 (UTC)

They aren't classifying tiling patterns, they are classifying Pentagons that are able to make a pattern, which is a *tiny* subset of the Pentagons that exist...Naraht (talk)

On silly games, I agree it is frustrating, but the rules are set. Fortunately its mainly types 1 and 2 that have the most degrees of freedoms and topological "families". I wouldn't be against moving the topic to its own article, like Monohedral tilings of convex pentagons to help reduce confusion. Tom Ruen (talk) 03:24, 12 September 2015 (UTC)