Rotatopes, towers, and arrays

Discussion of shapes with curves and holes in various dimensions.

Rotatopes, towers, and arrays

Postby Marek14 » Fri Aug 19, 2005 7:23 pm

The rotatope page here, among other things, shows rotatopes in a "progression mode" - i.e. what would a (n-1)-dimensional observer see should a n-dimensional rotatope pass through his or her dimension of choice.
This is mathematically equivalent to giving a fixed value of a to one of the coordinates in said rotatope's equation, and looking at the series of shapes it makes as the a slides from -1 to 1.

There's nothing, however, that would prevent us from, say, attaching fixed values to more than one coordinates at time, and showing the result in a two- or three-dimensional array. In fact, this is very nice way of showing the relationships between various pairs of coordinates.

First of all, let's look at the "tower" representations (1+2 dimensions) of 3D rotatopes:

CUBE has a single tower. Instead of slicing from -1 or 1, however, we start from 0 and build symmetrically to both sides of it. The 0-slice is a square, and as we go outwards, it stays that way.

CYLINDER has two towers. One of them starts as a circle, and stays identical as we move outwards. The second tower is more interesting, as it starts as square, and with us going outwards, one dimension of the square is steadily shrinking, while the others keeps its length. When we arrive to 1 or -1, we have only a line.

DOME has two towers as well. One of them starts as a square, but as we go outwards, BOTH of its dimensions shrink, until it becomes a point. The second tower starts as a circle, which shrinks along one of its axes until it becomes a line. The intermediate stages of this shrinking look like circles with smaller or larger parts at the top and bottom "cut off", and this abrasion continues until finally a single diameter is all that's left.

SPHERE has only a single tower. It starts as a circle, and shrinks into a point.

Now, for the 4D rotatopes. Here, I will show both tower representations in (1+3) dimensions, and array representation in (2+2) dimensions.

TESSERACT has a single tower which starts with a cube and continues in identical manner outwards. It has also one array representation.
In this representation, we start with a square. We go in four cardinal directions, filling them with identical squares. Finally, we make those two lines into a complete square, filling it as well with identical squares.

CUBINDER has two different towers - one is made of identical cylinders, the other has a cube in the middle, with one direction shrinking outwards, until it becomes a mere square.
There are three different arrays for it:
The first array is formed from the same identical squares as the tesseract one. The difference is that instead of filling a square with these squares, we fill a mere circle. When we go through the rows of this array, we find that there is a square/cube/square progression - a cubinder tower. The same is found when we go through columns.
The second array is a square of squares. This time, though, they are not identical - their width stays the same, but the height decreases from the middle upwards and downwards. Going from the top row, we find the square/cube/square progression again, but going from left to right we get identical cylinder every time.
The third array is simply a square array of identical circles. In this array, there are identical cylinders in all rows and columns.

DOMINDER has three towers and four arrays:
First tower has a cube in the middle. Two of its dimensions will shrink, leading to line/cube/line progression.
Second tower has a cylinder in the middle. One of its circular dimensions shrinks, until it becomes a square.
Third tower has a dome in the middle, and it's made up of identical domes.
Array 1 is circular. It has a middle row of identical squares and middle column of squares whose height shrinks until they become lines. The rows read line/cube/line, the columns read square/cylinder/square.
Array 2 is square. The middle row are identical squares, the middle column are squares shrinking in both dimensions to the point. Rows read line/cube/line, columns read dome/dome/dome
Array 3 is square. The middle row are squares whose height shrinks. So is the middle column. In fact, the height of any square in the array is given by 1-max(|x|,|y|). All the squares on the outside are shrunk into lines. Both rows and columns read square/cylinder/square
Array 4 is square. The middle row are identical circles. The middle column are circles whose height shrinks into line. Rows read square/cylinder/line, columns read dome/dome/dome.

DUOCYLINDER has one tower and two arrays.
The tower has cylinder in the middle whose height shrinks until it becomes a circle.
Array 1 is circular, made of identical circles. Rows and columns read circle/cylinder/circle.
Array 2 is square. The middle row is made of squares whose width shrinks. The middle column is made of squares whose height shrinks. The array is filled by rectangles of various sides. The edges are made of lines, and the vertices of the array are mere points. Again, both rows and columns read circle/cylinder/circle.

TRIDOME has two towers and two arrays.
Tower 1 starts with cube, and all three of its dimensions shrink into a point.
Tower 2 starts with a dome which shrinks along its height into a square.
Array 1 is circular. The middle row is made of identical squares. The middle column is made of squares whose both dimensions shrink into point. Rows read point/cube/point, columns read square/dome/square.
Array 2 is square. The middle row is made of circles whose height shrinks. The middle column is too. Once again, the height of circle in general point is 1-max(|x|,|y|). I will say that the height changes in SQUARE LAYERS, since this behaviour is quite common. Both rows and columns read square/dome/square.

SPHERINDER has two towers and two arrays.
Tower 1 starts with a cylinder whose diameter shrinks into a line.
Tower 2 is made of identical spheres.
Array 1 is circular. The middle row and middle column are made of squares whose height shrinks. The height of square changes with distance from center, the circumference of the circle is made of lines. Rows and columns read line/cylinder/line.
Array 2 is square. The middle row are identical circles, the middle column are circles that shrink into a point. Rows read line/cylinder/line, columns read sphere/sphere/sphere.

LONGDOME has two towers and four arrays.
Tower 1 starts with a cylinder. Its height and one of its circular dimensions shrink, so it becomes a line at the end.
Tower 2 starts with a dome. One of its horizontal dimensions shrinks so it becomes a circle at the end.
Array 1 is a circular array of squares whose height varies with the row and width with the column. This finally gives those of us with experience in 4D carpentry a way how to construct longdome: Take a duocylinder, orient it until it has square shadow in certain two dimensions, then saw off everything that is not contained within a circle in the other two dimensions. Easy! Both row and columns of this array read line/cylinder/line.
Array 2 is circular again. This time it's filled with circles whose height shrinks with row. Rows read line/cylinder/line, columns read circle/dome/circle.
Array 3 is square. It's filled with squares whose height shrinks in square layers, and whose width shrinks with row only. Once again, the row read line/cylinder/line, and columns read circle/dome/circle.
Array 4 is square, and contains circles. Height of the circles varies with row, and width with column. The circles in this array are cut from above and below, or from sides, all this in various sizes. Both rows and columns read circle/dome/circle.

SPHERIDOME has 3 towers and 4 arrays.
Tower 1 starts with a cylinder which shrinks into a point.
Tower 2 starts with a dome whose height and one horizontal dimension shrink until only a line is left.
Tower 3 starts with a sphere whose height shrinks until it becomes a circle.
Array 1 is circular, made of squares. Height of squares shrinks with the row, while their width shrinks with the distance from the center. Rows read point/cylinder/point while columns read line/dome/line.
Array 2 is circular, too. It's made of circles whose diameter shrinks with the row. Rows read point/cylinder/point, while columns read circle/sphere/circle.
Array 3 is circular, too. It's made of circles whose height shrinks with the distance from the center. Both rows and columns read line/dome/line.
Array 4 is square and it's made of circles whose height shrinks with row, and width shrinks in square layers. Rows read line/dome/line, columns read circle/sphere/circle.
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Postby wendy » Sat Aug 20, 2005 11:17 am

you trying to do sections and lace-cities?

It's pretty easy to work this sort of stuff from your notation, i should imagine.

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the dream we dream together is reality.

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Postby Marek14 » Sun Aug 21, 2005 7:10 am

Yes, it's easy.
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