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The Generalized Dual Method, or GDM, in two dimensions, is named after a
peculiar duality between:
1. An arrangement of lines in the plane.
2. A tiling by zonogons (centrally-symmetrical polygons).
The Penrose tiling of 72- and 36-degree rhombs may be obtained using the GDM,
for instance.
The "duality" arises because if two lines in the arrangement intersect, a rhomb
arises in the tiling. Every open space between lines, bounded or unbounded,
corresponds to a vertex in the tiling; and every segment or ray cut out by the
lines corresponds to an edge in the tiling.
If three lines intersect in one point, a hexagon arises; if four lines, an
octagon. And so on. These may or may not be regular.
One could pass a set of lines in general position to the GDM, no two lines
parallel; only rhombs would arise.
More typically, one explores symmetry, and makes an arrangement of lines using
"n" (an integer) subsets, each subset parallel to one of the sides of a regular
n-gon. Each subset is composed of, say, "k" (an integer) parallel lines.
Typically, they are evenly-spaced. But they need not be evenly-spaced.
Research into such tilings often restricts them to rhombic tilings; it is
ensured that no more than two lines intersect at any one point. I myself often
prefer to explore the opposite: I force as many lines as possible to intersect
at as many points as possible.
If n=5, each subset is parallel to one of the sides of a regular pentagon, then;
I begin with an arrangement of five lines, arranged so they form a kind of star
pentagon, a pentagram (but the lines extend beyond the pentagram to infinity).
I find the ten points of intersection, and I add another set of five lines, at
distances such that the initial ten points of intersection are all hit again by
my new lines. I poll this new arrangement for all its points of intersection,
and add more lines, forcing more intersections. And then, more.
When five lines intersect at one point, then, a regular decagon arises. And if
five lines intersect in many points, many decagons arise ...
Post a reply to this message
Attachments:
Download 'higher_zonogons.jpg' (132 KB)
Preview of image 'higher_zonogons.jpg'
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"Russell Towle" <rto### [at] inreach com> wrote:
> The Generalized Dual Method, or GDM, in two dimensions, is named after a
> peculiar duality between:
>
> 1. An arrangement of lines in the plane.
>
> 2. A tiling by zonogons (centrally-symmetrical polygons).
>
> The Penrose tiling of 72- and 36-degree rhombs may be obtained using the GDM,
> for instance.
>
> The "duality" arises because if two lines in the arrangement intersect, a rhomb
> arises in the tiling. Every open space between lines, bounded or unbounded,
> corresponds to a vertex in the tiling; and every segment or ray cut out by the
> lines corresponds to an edge in the tiling.
>
> If three lines intersect in one point, a hexagon arises; if four lines, an
> octagon. And so on. These may or may not be regular.
>
> One could pass a set of lines in general position to the GDM, no two lines
> parallel; only rhombs would arise.
>
> More typically, one explores symmetry, and makes an arrangement of lines using
> "n" (an integer) subsets, each subset parallel to one of the sides of a regular
> n-gon. Each subset is composed of, say, "k" (an integer) parallel lines.
> Typically, they are evenly-spaced. But they need not be evenly-spaced.
>
> Research into such tilings often restricts them to rhombic tilings; it is
> ensured that no more than two lines intersect at any one point. I myself often
> prefer to explore the opposite: I force as many lines as possible to intersect
> at as many points as possible.
>
> If n=5, each subset is parallel to one of the sides of a regular pentagon, then;
> I begin with an arrangement of five lines, arranged so they form a kind of star
> pentagon, a pentagram (but the lines extend beyond the pentagram to infinity).
> I find the ten points of intersection, and I add another set of five lines, at
> distances such that the initial ten points of intersection are all hit again by
> my new lines. I poll this new arrangement for all its points of intersection,
> and add more lines, forcing more intersections. And then, more.
>
> When five lines intersect at one point, then, a regular decagon arises. And if
> five lines intersect in many points, many decagons arise ...
Ideas well explained, should be enough. I think I need more (being slow) or need
to read that 4 more times. Starting to find something to read now. I really
think pov needs a gdm.inc include file. (and will once I understand this)
Post a reply to this message
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"alphaQuad" <alp### [at] earthlinknet> wrote:
> Ideas well explained, should be enough. I think I need more (being slow) or need
> to read that 4 more times. Starting to find something to read now. I really
> think pov needs a gdm.inc include file. (and will once I understand this)
Here is an image showing, on the left, a line arrangment, on the right, its
generalized-dual zonogonal tiling. Here, n=9. Since there no more than two
lines ever intersect in one point, the only zonogons are rhombs.
Notice that in the line arrangement, there is a central hollow area bounded by
nine segments. In the tiling, there is a central vertex, where nine edges meet.
The right-most point of intersection in the line arrangement corresponds to the
right-most rhomb in the tiling.
There are thirty-six points of intersection in the arrangement, and thirty-six
rhombs in the tiling.
There are eighteen unbounded regions between lines around the arrangement; and
eighteen vertices are on the boundary of the tiling.
Post a reply to this message
Attachments:
Download 'bijection.jpg' (34 KB)
Preview of image 'bijection.jpg'
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