All Squares Are Rectangles

[Suyay Escarsega]

Iam trying to write a method which will take my input dimensions (9' x 8' 8") and min/max size (1' x 3', 2', 4', etc..) and generate a random pattern of squares and rectangles to fill the wall. I tried doing this by hand, but I'm just not happy with the layout that I got, and it takes about 35 minutes each time I want to 'randomize' the layout.

One solution is to start with x*y squares and randomly merge squares together to form rectangles. You'll want to give differing weights to different size squares to keep the algorithm from just ending up with loads of tiny rectangles (i.e. large rectangles should probably have a higher chance of being picked for merging until they get too big).

It might be interesting to parametrize the generation of points, and then make a genetic algorithm. The fitness function will be how much you like the arrangement - it would draw hundreds of arrangements for you, and you would rate them on a scale of 1-10. It would then take the best ones and tweak those, and repeat until you get an arrangement you really like.

This actually sounds more like an old school random square painting demo, circa 8-bit computing days, especially if you don't mind overlaps. But if you want to be especially geeky, create random squares and solve for the packing problem.

I would generate everything in a spiral slowly going in. If at any point you reach a point where your solution is proven to be 'unsolvable' (IE, can't put any squares in the remaining middle to satisfy the constraints), go to an earlier draft and change some square until you find a happy solution.

Then choose a random vertex, find polygon lines within (inclusive) 2 X maxside of the line.Find x values of all vertical lines and y values of all horizontal lines. Create ratings for the "goodness" of choosing each x and y value, and equations to generate ratings for values in between the values. Goodness is measured as reducing number of lines in remaining polygon. Generate three options for each range of values between two x coordinates or two y coordinates, using pseudo-random generator. Rate and choose pairs of x and pair of y values on weighted average basis leaning towards good options. Apply new rectangle to list by cutting its shape from the poly array and adding rectangle coordinates to the dest array.

Question does not state a minimum side parameter. But if one is needed, algorithm should (upon hitting a hitch with a gap being too small) not include too small candidates in selection lists (whic will occasionally make them empty) and deselect a number of the surrounding rectangles in a certain radius of the problem with size and perform new regeneration attempts of that area, and hopefully the problem area, until the criteria are met. Recursion can remove progressively larger areas if a smaller relaying of tiles fails.

The similarities in the properties of a square and a rectangle can help us draw a conclusion to the query "Is square a rectangle". In Geometry, we have learned about different types of shapes such as square, rectangle, cylinder, rhombus, cuboid, cube, cone, parallelogram, and so on. Many of these shapes share certain common properties. Square and rectangle are examples of such two-dimensional shapes. They both fall under the category of quadrilaterals.

Square: A square is a two-dimensional plane figure with four equal sides, four interior right angles, and four corners. In other words, a square is a quadrilateral or a 4-sided polygon. All the angles are of equal measure, therefore it is considered as an equiangular quadrilateral.

Rectangle: A rectangle is a two-dimensional figure with four sides, four interior right angles, and four corners. The opposites sides of a rectangle are equal. A rectangle has four angles, with each angle measuring 90. Similar to a square, a rectangle is also referred to as an equiangular quadrilateral.

Since both a square and a rectangle have an equal number of sides, i.e., 4, we thus can conclude that both square and rectangle are quadrilaterals. We can observe the similarities in the properties of both a square and a rectangle in the following section.

Both a square and a rectangle have certain special properties that distinguish them from a general quadrilateral. We can draw the comparison of the requisite properties of both the shapes from the following table and conclude whether a square has all the properties that define a rectangle.

From the comparison drawn above for the common properties shared between a square and a rectangle, we observe that a square has all the properties that define a rectangle, which makes them alike in a certain manner. This means that a square can also be referred to as a type of rectangle.

Example 2: There is a park near Sam's house. The length of the sides of the park is 300 meters, 400 meters, 300 meters, and 400 meters. Determine whether the park is a square or a rectangle.

The length of each side of the park is given as 300 meters, 400 meters, 300 meters, and 400 meters. Now, we know that all four sides of a square are equal. Hence, this is not a square. Given that a rectangle has opposite sides equal, we can see that the given dimensions are of a rectangle, and not a square.

A square is both a rectangle and a rhombus because it possesses the properties of rectangles and rhombuses. If you are also looking for the answer to the question - is a rhombus a rectangle, the answer is no.

A square is a special type of rectangle because it possesses all the properties of a rectangle. Similar to a rectangle, a square has interior angles which measure 90 each. opposite sides that are parallel and equal.

A square is both a rectangle and a rhombus because it possesses the properties of rectangles and rhombuses. If you are also looking for the answer to the question - is a rhombus a rectangle, the answer is no.\u00a0

A square is a special type of rectangle because it possesses all the properties of a rectangle. Similar to a rectangle, a square has interior angles which measure 90\u00ba\u00a0each. opposite sides that are parallel and equal.

Any given square is always a rectangle, but a rectangle isn't necessarily a square, so squares and rectangles have a _ relationship. I've been noticing this sort of thing everywhere ever since I noticed that I didn't know a good word for it. I've been calling it a container relationship because one class of things is contained within a larger class of things. However, a "container" relationship doesn't sound very good and doesn't really convey the meaning on its own. A more elegant word would be helpful.

In linguistics, a hyponym is a word or phrase whose semantic field is included within that of another word, its hypernym (sometimes spelled hyperonym outside of the natural language processing community). In simpler terms, a hyponym shares a type-of relationship with its hypernym. For example, "pigeon", "crow", "eagle" and "seagull" are all hyponyms of "bird" (their hypernym); which, in turn, is a hyponym of "animal".

A rectangle definitely does not "contain" a square, in the sense of the original question: that would be the wrong thing to say to mean that a square is a type of rectangle. When you say "a rectangle contains a square" you mean "has a" not "is a".

It's also "obscure" to think of a "square" as a subset of a "rectangle". The set of squares might be a subset of the set of rectangles, but "square" and "rectangle" in this context are types (of shapes), not sets.

You can consider the set of all squares and the set of all rectangles and how they overlap with one another. In mathematical jargon we would say Squares are a proper subset of rectangles. or Rectangles are a proper superset of squares.

You can also consider the meanings of square and rectangle and how they relate to one another. The meaning of square can be expressed in terms of being a rectangle with additional restriction: A square is an equilateral rectangle. So the relationship could be expressed as: A square is a kind of rectangle. There are a wide range of other ways of expressing kind of. Depending on the direction, the relationship itself is called specialization or generalization.

Most people are more comfortable expressing things in terms of subsets or specialization than they are in terms of supersets or generalization. So unless you are in a technical context, or the direction is very important to your meaning, it's best to stick to expressing these relationships in terms of specialization.

I had the same original question as you. Perhaps "mutual inclusion" or "mutually inclusive."I wad having an argument with someone when I said I liked to ride motorcycles because they are fun, and she replied "oh, it's a game to you?" Games are fun, leading her to this conclusion. But, not all fun derives from games. Fun can come from other forms. I thought about the rectangle/square relation, and it amazed me that I didn't know the term for their relationship. After reading the responses here, saying that two things are not mutually inclusive seems like a reasonable phrase.

There are the same number of rectangles as squares for the reason you mention, even though the set of squares is a proper subset of the set of rectangles. It's no stranger than the fact that there are the same number of even integers as there are integers. Every infinite set can be placed in one-to-one correspondence with a proper subset of itself.

It depends on what you mean by "more", a "square", and a "rectangle". For the sake of this discussion, I am going to assume that a square is a regular quadrilateral, and that a rectangle is a equiangular quadrilateral. I will be working in Euclidean space, which is coordinate-free (i.e. I do not impose a Cartesian coordinate system upon this space; the answer of both Gerry Myerson seems to implicitly impose this condition, while csch2's answer does not, though it still seems to think of rectangles and squares as living in Cartesian space, rather than Euclidean space). In this setting, a square corresponds to a single parameter (its side-length), and a rectangle corresponds to two parameters (the lengths of two adjacent sides).

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