Brick product (InstanceTopic, 3)

From Hi.gher. Space

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Revision as of 16:28, 4 January 2012

The brick product is a very generalized operation, written as Q = P{A₁, A₂, ..., A_n}. P is called the operator and A_i are called the operands. Their values must satisfy the following constraints:

P must be a brick.
The dimensionality of P must equal n.

The resultant shape Q will have the following properties:

The dimensionality of Q will be the sum of the dimensionalities of A_i.
Q will be a brick if and only if all A_i are bricks.
Q will be convex if and only if both P and all A_i are convex.

Definition

For each point x = (x₁, x₂, ..., x_n) in P, the brick symmetry guarantees that (-x₁, x₂, ..., x_n) is also in P. This means the line segment between these two points has length 2x₁. In fact, for any 1 ≤ i ≤ n, a line segment can be constructed between (x₁, ..., x_i, ..., x_n) and (x₁, ..., -x_i, ..., x_n) with both endpoints in P, by definition of brick symmetry of P, each with length 2x_i. Collect these lengths into an array l(x) = (l₁, l₂, ..., l_n) = (2x₁, 2x₂, ..., 2x_n).

Now, for each point x, we form the Cartesian product A₁|l₁ × A₂|l₂ × ... × A_n|l_n, where the notation A_i|l_i means the operand A_i scaled by the length l_i.

Then Q = P{A₁, A₂, ..., A_n} is the union over all x in P of the surfaces of the Cartesian products A₁|l₁ × A₂|l₂ × ... × A_n|l_n.

As a concise expression:

Q = ⋃{ surface( ∏{ A_i|2x_i | 1 ≤ i ≤ n } ) | x ∈ P }

where ⋃ represents the union of all members of a set, ∏ represents the Cartesian product of all members of a set, and surface(X) is a function mapping X to its preceding frame, i.e. reduces the net space of X by one while keeping the bounding space the same.

Specializations

When P is the point, the operation is trivial, takes no operands, and the result by the definition above is an empty set; some uses may prefer to define Point{} = Point.
When P is the digon, the operation is simply scales its operand by the digon's length. If the length of the digon is 1, the operation becomes the identity operation.
When P is a hypercube of side length 1, the operation is the Cartesian product.
When P is a cross polytope of diameter 1, the operation is the tegum product.
When P is a hypersphere of diameter 1, the operation is the crind product.
When all A_i are equal, the operation produces a powertope (A_i^P, read "the P of A_i").
- When each A_i is also equal to P, the operation produces a tetrate (the P-al n-tetrate).

@@ Line 9: / Line 9: @@
 == Definition ==
+For each point ''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) in P, the brick symmetry guarantees that (-''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) is also in P. This means the line segment between these two points has length 2''x''<sub>1</sub>. In fact, for any 1 ≤ ''i'' ≤ ''n'', a line segment can be constructed between (''x''<sub>1</sub>, ..., ''x''<sub>''i''</sub>, ..., ''x''<sub>''n''</sub>) and (''x''<sub>1</sub>, ..., -''x''<sub>''i''</sub>, ..., ''x''<sub>''n''</sub>) with both endpoints in P, by definition of brick symmetry of P, each with length 2''x''<sub>''i''</sub>. Collect these lengths into an array ''l''(''x'') = (''l''<sub>1</sub>, ''l''<sub>2</sub>, ..., ''l''<sub>''n''</sub>) = (2''x''<sub>1</sub>, 2''x''<sub>2</sub>, ..., 2''x''<sub>''n''</sub>).
-For each point ''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)/2 in P, the brick symmetry guarantees that (-''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)/2 is also in P. This means the line segment between these two points has length ''x''<sub>1</sub>. In fact, for any 1 ≤ ''i'' ≤ ''n'', a line segment can be constructed between (''x''<sub>1</sub>, ..., ''x''<sub>''i''</sub>, ..., ''x''<sub>''n''</sub>)/2 and (''x''<sub>1</sub>, ..., -''x''<sub>''i''</sub>, ..., ''x''<sub>''n''</sub>)/2 with both endpoints in P, by definition of brick symmetry of P, each with length ''x''<sub>''i''</sub>.
+Now, for each point ''x'', we form the [[Cartesian product]] A<sub>1</sub>|''l''<sub>1</sub> × A<sub>2</sub>|''l''<sub>2</sub> × ... × A<sub>''n''</sub>|''l''<sub>''n''</sub>, where the notation A<sub>''i''</sub>|''l''<sub>''i''</sub> means the operand A<sub>''i''</sub> scaled by the length ''l''<sub>''i''</sub>.
-Now, for each point ''x'', we form the [[Cartesian product]] (''x''<sub>1</sub>A<sub>1</sub>) × (''x''<sub>2</sub>A<sub>2</sub>) × ... × (''x''<sub>''n''</sub>A<sub>''n''</sub>), where the notation ''x''<sub>''i''</sub>A<sub>''i''</sub> means the operand A<sub>''i''</sub> scaled by ''x''<sub>''i''</sub>.
+Then Q = P{A<sub>1</sub>, A<sub>2</sub>, ..., A<sub>''n''</sub>} is the union over all ''x'' in P of the surfaces of the Cartesian products A<sub>1</sub>|''l''<sub>1</sub> × A<sub>2</sub>|''l''<sub>2</sub> × ... × A<sub>''n''</sub>|''l''<sub>''n''</sub>.
-Then Q = P{A<sub>1</sub>, A<sub>2</sub>, ..., A<sub>''n''</sub>} is the union over all ''x'' in P of the surfaces of the Cartesian products (''x''<sub>1</sub>A<sub>1</sub>) × (''x''<sub>2</sub>A<sub>2</sub>) × ... × (''x''<sub>''n''</sub>A<sub>''n''</sub>).
 As a concise expression:
 <blockquote style='font-size: 200%;'>
-Q = ⋃{ <span style='font-size: 80%;'>surface( <span style='font-size: 80%;'>∏{ <span style='font-size: 80%;'>''x''<sub>''i''</sub>A<sub>''i''</sub></span> | <span style='font-size: 80%;'>1 ≤ ''i'' ≤ ''n''</span> }</span> )</span> | <span style='font-size: 80%;'>''x'' ∈ P</span> }
+Q = ⋃{ <span style='font-size: 80%;'>surface( <span style='font-size: 80%;'>∏{ <span style='font-size: 80%;'>A<sub>''i''</sub>|2''x''<sub>''i''</sub></span> | <span style='font-size: 80%;'>1 ≤ ''i'' ≤ ''n''</span> }</span> )</span> | <span style='font-size: 80%;'>''x'' ∈ P</span> }
 </blockquote>
 where ⋃ represents the union of all members of a set, ∏ represents the Cartesian product of all members of a set, and surface(X) is a function mapping X to its preceding [[frame]], i.e. reduces the net space of X by one while keeping the bounding space the same.
+== Specializations ==
+*When P is the [[point]], the operation is trivial, takes no operands, and the result by the definition above is an empty set; some uses may prefer to define Point{} = Point.
+*When P is the [[digon]], the operation is simply scales its operand by the digon's length. If the length of the digon is 1, the operation becomes the identity operation.
+*When P is a [[hypercube]] of side length 1, the operation is the Cartesian product.
+*When P is a [[cross polytope]] of diameter 1, the operation is the [[tegum product]].
+*When P is a [[hypersphere]] of diameter 1, the operation is the [[crind product]].
+*When all A<sub>''i''</sub> are equal, the operation produces a [[powertope]] (A<sub>''i''</sub><sup>P</sup>, read "the P of A<sub>''i''</sub>").
+**When each A<sub>''i''</sub> is also equal to P, the operation produces a ''tetrate'' (the P-al ''n''-tetrate).