ManyFolds in Variety

This is an example of 'polygrammic arrangement'.
Two regular ({12/4}-) dodecagrammic circular arrangements and these in combination:


Displayed at the 6th annual convention held by Kansai branch of Japan Origami Academic Society (JOAS) - Origami Tanteidan Kansai Circle -, April 30 - May 1 (Yesterday), 2005 at Kobe, Japan.

Material: 12 sheets of square origami papers for each, without using adhesives.

Date: In April 2005

Photo date: May 2, 2005

Click each image to see the larger picture.

Helical folding structure (Convolution) (4)

A regular 'polygrammic' helical folding structure
in convolution

[Outline of the explanation in Japanese added to the drawing is :]

In case of θ = pπ/n and φ = (q-1) θ,
where n, p, q are natural numbers satisfying
p, q ≧ 2, pq ≦ n/2,
n and p are mutually prime natural numbers.
The drawing is the case at (n, p, q) = (15, 2, 3): a regular pentadecagrammic helical folding structure in convolution,
that consists of the union of three pentadecagrams:
{15/2}, {15/4} and {15/6}, as presented in mathematical notation.

Helical folding structure (Convolution) (4')

[Outline of the explanation in Japanese added to the drawing is :]

In case of φ = θ = pπ/n,
where n and p are mutually prime natural numbers
satisfying 2 ≦ p ≦ n/4.
The drawing is in case at (n, p) = (16, 3), where the edges form the union of two heptadecagrams {16/3} and {16/6},
as presented in mathematical notation.

See the article: Helical piled up structure over a hexadecagram.

Click each image to see the larger picture.

Helical folding structure (Convolution) (3)

A regular polygonal helical folding structure in convolution

[Outline of the explanation in Japanese added to the drawing is :]

In case of θ = π/n, φ = (m - 1) θ,
where m and n are natural numbers satisfying 2 ≦ m ≦ n/2.
The drawing is the case at (n, m) = (12, 4),
a regular dodecagonal helical folding.

Helical folding structure (Convolution) (3) - Example #1 -

In case of θ = π = φ/5 at n = 5 and m = 2
A regular pentagonal helical folding structure in convolution.

Helical folding structure (Convolution) (3) - Example #2 -

In case of θ = π/8 and φ = π/4 at n = 8 and m = 3
A regular octagonal helical folding structure in convolution.

Click each image to see the larger picture.

Helical folding structure (Convolution) (2)

[Outline of the explanation in Japanese added to the drawing is :]

Generated by given two angles (positive real numbers)
θ and φ such that φ = (k-1) θ,
where k is a natural number satisfying 0 < k θ ≦ π/2 (rad).
Namely, in case of (1) for φ is just a positive multiple of θ.
Note that it is possible not to be contained in the area of a (regular) polygon: if the strip of infinite length will be folded, then countable infinite number of all vertices set on that strip for helical folding should be placed dense in the circumscribing circle.

See the article: Helical piled up structure over an annulus.

Helical folding structure (Convolution) (2')

In case of k = 2 at (2); φ = θ and 0 < 2 θ ≦ π/2 (rad).

See also Helical piled up structure over an annulus.

In the following series of articles (from #1 to #4), each image has been revised from his original handwritten drawing on a basic single layer strip with helical folding patterns, the folded shapes with explanation (in Japanese) in mathematical notation and corresponding trigonometric equations.
These had been prepared as his subscripts for the article to appear on the quarterly ORU #6 (1994 Autumn) and the Second International Meeting of Origami Science and Scientific Origami, held at Seian University of Art and Design, Otsu, Japan, November 29-December 2, 1994.
[Though the diagrams did not appeared in the magazine nor be used at his presentation in the meeting, some people received copies or modified articles of .pdf files in respective contacts with him or by mail/e-mail.]

Click the image to see the larger picture.

Helical folding structure (Convolution) (1)

[Subtitle in Japanese of the above drawing is:]

Generated by given two angles (positive real numbers)
θ and φ satisfying 0 < θ + φ ≦ π/2 (rad).