K-PATTERNS
Back to GEOMETRY GARRET
Alan H. Schoen
schoenah@gmail.com
Comments are welcome!
D. K-PATTERNS: IMAGES DERIVED FROM PARTIAL SUMS OF POWER RESIDUES
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PRELIMINARIES
The kth vertex (k = 0,1,2,...) of a K-pattern is located at
j0= non-negative integer.
n is called the modulus,
α the exponent, and
σ the [loop] step.
instead of having consecutive vertices rk joined by a unit vector,
(a) every computed unit vector is divided into two collinear
half-unit vectors (which are not plotted),
and then
(b) the midpoints of consecutive half-unit vectors are joined
by a line segment.
All of the 54 patterns in the 6 x 9 table of images (top) are rounded.
that assigns different colors to the various line segments
in a K-pattern with reflection symmetry
so that it will display color symmetry.
Summary report : a contributed paper delivered in Section 03
at ICM 86 in Berkeley, on August 8, 1986, entitled
"An efficient algorithm for locating lines of reflection in K-patterns —
computer graphics patterns derived from partial sums of power residues"
CATALOG OF SELECTED IMAGES (1986)
the first is the exponent α,
the second is the [loop] step σ,
the third is the modulus n.
Part 2
Part 3
Part 4
Part 5
Part 6
Part 7
Part 8
From the great variety of families of K-patterns,
I have selected below some examples of
CAT'S CRADLE and
DECORATED CYCLOID
patterns.
α = p, and
σ = odd integer.
It contains a single horizontal unit vector.
If σ ≠ k p (k = odd integer), the pattern is non-trivial.
It contains 2p2 unit vectors and has d2 symmetry.
One line of reflection is horizontal and the other is vertical.
There are p(p − 1)/2 non-trivial patterns,
each composed of two interlaced
sub-patterns L and R, which are
congruent and have dp symmetry.
Each of these sub-patterns is rotated about its center
through the angle π/p with respect to the other.
L and R each contain p congruent replicas of a pattern motif M,
a d1-symmetric chain of p − 1 unit vectors, centered at angular positions
that are uniformly distributed around the respective centers of L and R.
Each replica of M in L(R) is joined at either end
by a horizontal unit vector
to a replica of M in R(L).
Consecutive instances of these linking vectors have opposite directions.
Below is a CAT'S CRADLE pattern for
(p, σ, j0) = (11,1,0).
Both the set Sred of eleven red instances of the ten-vector motif M and
the set Sgreen of eleven green instances of M
have d11 symmetry.
(Both the orange and blue dashed regular 11-gons also have d11 symmetry.)
n = p3 (p = odd prime),
α = p, and
σ = even integer.
If σ = k p (k = even integer), the pattern is trivial.
It contains a single horizontal unit vector.
If σ ≠ k p (k = even integer), the pattern is composed of
two interlaced sub-patterns A and B, which are congruent.
A and B each contain p congruent replicas of a pattern motif M,
which is a d1-symmetric chain of p − 1 unit vectors.
Each replica of M in A(B) is joined at either end
by a horizontal unit vector to a replica of M in B(A).
Consecutive instances of these linking vectors have the same directions.
As is illustrated below, the vertices at the centers of every instance of M
lie on the same cycloid — hence the family name, 'DECORATED CYCLOID'.
Below is a DECORATED CYCLOID frieze pattern for (p, σ, j0) = (11,2,0).
Just as in the CAT'S CRADLE for
(p, σ, j0) = (11,1,0) shown above,
the [congruent] red and green instances of the ten-vector pattern motif M
each have d1 symmetry.
The modern symbol for the symmetry type (called 'sidle') of the entire frieze,
which has two parallel lines of reflection, is
.gif){kind=small}
(cf. p. 68 of 'The SYMMETRIES of THINGS',
by Conway, Burgiel, and Goodman-Strauss,
A.K. Peters, Ltd, 2008).
there exists a simpler unlinked centro-symmetric pattern of d2p symmetry
based on the same motif M.
In these patterns, therre are no linking vectors,
and the motifs in each of the two interlaced motif subsets
are all at the same radial distance from the center of the pattern.
The central angles of consecutive instances of the motif differ by Δθ = σ π/p (mod 2π).
Below is the complete set of three non-trivial
CAT'S CRADLE patterns for p = 3 (right),
together with their associated unlinked d6 patterns (left).
σ = 1
Δθ = 60°
σ = 5
Δθ = − 60°
σ = 7
Δθ = 60°
Below is the complete set of twenty-one non-trivial
CAT'S CRADLE patterns for p = 7 (right),
together with their associated unlinked d14 patterns (left).
σ = 1
Δθ ≅ 25.7°
σ = 3 Δθ ≅ 77.1°
σ = 5 Δθ ≅ 128.6°
σ = 9 Δθ ≅ − 128.6°
σ = 11 Δθ ≅ − 77.1°
σ = 13 Δθ ≅ − 25.7°
σ = 15 Δθ ≅ 25.7°
σ = 17 Δθ ≅ 77.1°
σ = 19 Δθ ≅ 128.6°
σ = 23 Δθ ≅ − 128.6°
σ = 25 Δθ ≅ − 77.1°
σ = 27 - Δθ ≅ − 25.7°
σ = 29 Δθ ≅ 25.7°
σ = 31 Δθ ≅ 77.1°
σ = 33 Δθ ≅ 128.6°
σ = 37 Δθ ≅ − 128.6°
σ = 39 Δθ ≅ − 77.1°
σ = 41 Δθ ≅ − 25.7°
σ = 43 Δθ ≅ 25.7°
σ = 45 Δθ ≅ 77.1°
σ = 47 Δθ ≅ 128.6°
The complete set of 10 non-trivial patterns of CAT'S CRADLE_5 (p = 5)
corresponds to σ = {1, 3, 7, 9, 11, 13, 17, 19, 21, 23}.
The entire ten-pattern sequence is displayed eight times.
The pattern for σ50k+j (j, k = 1, 2, 3,...) is identical to the pattern for σj.
To animate the sequence, hold down the 'PageDown' key.
The complete set of 21 non-trivial patterns of CAT'S CRADLE_7 (p = 7)
corresponds to σ = {1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47}.
The pattern for σ98k+j (j, k = 1, 2, 3,...) is identical to the pattern for σj.
To animate the sequence, hold down the 'PageDown' key.
The complete set of 78 non-trivial patterns of CAT'S CRADLE_13 (p = 13)
corresponds to σ = {1, 3, 5, 7, 9, 11, 15, 17, ..., 165, 167}.
To animate the sequence, hold down the 'PageDown' key.
The first 128 patterns of CAT'S CRADLE_17 (p = 17)
correspond to σ = {1, 3, 5, ..., 253, 255}.
Included are 7 instances of the trivial pattern.
(The complete set contains 136 distinct non-trival patterns.)
To animate the sequence, hold down the 'PageDown' key.
The first 48 patterns of CAT'S CRADLE_47 (p = 47)
correspond to σ = {1, 3, ..., 93}.
Included is one instance of the trivial pattern.
(The complete set contains 1081 non-trival patterns.)
To animate the sequence, hold down the 'PageDown' key.
You may need to
(a) adjust the ZOOM value to 100%
and then
(b) center the images on the screen.
The complete set of 10 non-trivial patterns for DECORATED CYCLOID_5 (p=5)
corresponds to σ = {2, 4, 6, 8, 12, 14, 16, 18, 22, 24}.
Each pattern is shown eight times.
The pattern for σ50k+j (j, k = 1, 2, 3,...) is identical to the pattern for σj.
To animate the sequence, hold down the 'PageDown' key.
The complete set of 21non-trivial patterns for DECORATED CYCLOID_7 (p=7)
corresponds to σ = {2, 4, 6, 8, 12, 14, 16, 18, 22, ..., 46, 48}.
The pattern for σ98k+j (j, k = 1, 2, 3,...) is identical to the pattern for σj.
To animate the sequence, hold down the 'PageDown' key.
The first 128 patterns of DECORATED CYCLOID_17 (p=17)
correspond to σ = {2, 4, ..., 256}.
Included are 7 instances of the trivial pattern.
(The complete set contains 136 distinct non-trival patterns.)
To animate the sequence, hold down the 'PageDown' key.
The first 140 patterns of DECORATED CYCLOID_47 (p=47)
correspond to σ = {2, 4, ..., 280}.
Included are 3 instances of the trivial pattern.
To animate the sequence, hold down the 'PageDown' key.