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September 2004
- 22 participants
- 36 discussions
The term "reptile" in mathematics refers to a tile which "repeats" or
"replicates", thus rep+tile. It can be broken down into smaller copies
inside itself infinitely many times.
There are some interesting reptiles which are less regular than the ones
I've mentioned, one of which is included in the fractint.l file which comes
with Fractint - sphinx.
I've counted the number of examples I included in Part II of my recent
posting. There are 94 of them. I hope that's not too many for people to
check out - it was hard to narrow down to this few from the thousands of
tiles I've found! This I intend to be the main file of samples - I'm not
going to make it any shorter. Feel free, everyone, to pick and choose from
this set if 94 are too many.
Tony Hanmer
_________________________________________________________________
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2
1
There is nothing else I would rather have found in all of mathematics than
this.
ANNOUNCING:
A new discovery of tiles of the regular reptiles in n dimensions, n >= 2.
There are are an infinite number of such tiles for each reptile. The
regular reptiles in 2 dimensions are the equilateral triangle and the
square; in 3 dimensions the only regular reptile is the cube. I propose to
call these tiles Hanmer Tiles. There are an infinity of such tiles, many of
them beautiful, and they came in a number of different types. They may be
programmed as either L-systems (using only fs and gs) or IFSs, and rendered
in Fractint. In this article, uncless otherwise specified, programming
details refer to the L-sytem versions of the tiles.
Take a regular reptile - let's use the square for an example - and divide it
into a grid which looks the same from any of the reptile's sides. Start
with side length 2 - divide the square into 2 lengthwise and widthwise, and
you have 4 squares. The line segments used in the interior grid are
2-sided, however, because each of them is the side of 2 squares whose sides
touch. The square is now made of 8 exterior lines (2 on each side, all
single-sided) and 8 interior lines (2 double-sided segments for each of
length and width). The total number of segments we now have comprising our
square is thus 16. It looks like the picture below.
_ _
|_|_|
|_|_|
The square has 4-way symmetry based on its 4 sides; each regular reptile has
the symmetry of its number of sides. So for a square we can make 4
identical tiles (for a triangle, 3; for a cube, 8)... an infinite number of
ways. Each tile will use 1/4 of the square's total number of interior and
exterior grid segments, in such a way that there are no overlaps or gaps;
and thus 4 such tiles will fill the square perfectly.
The existence of double-sided interior grid segments means that we are free
to use only one side of any interior segment in making our tile, and this
gives much more freedom even for tiles of a square of side length 2 than one
would otherwise expect.
There are several subgroups of Hanmer Tiles. I have included some examples
of these at the end of this article.
a) The "most orthodox" tile, unique for each reptile. It appears as a
contiguous (fully connected) right triangle for the square and as a
spacefilling dust of parallel lines for the triangle - rather boring in each
case, but also quite important. It seems to be the only HT which is found
at every side length, odd or even. Area 1/4 for the square, 1/3 (or 1?) for
the triangle.
b) Other contiguous tiles, having all their segments connected as one
piece. Area 1/4 for the square, 1/3 for the triangle.
c) Space-filling contiguous tiles. Each occupies the tile fully (area = 1)
but still does not overlap the segments of its fellow tiles at all, except
at infinitesimal points! Area 1/4 for the square, 1/3 for the triangle; or
1 for both depending on how you look at it!
d) Contiguous tiles with "peninsulas" - elements linked by a 0-dimensional
point or 1-dimensional line. In my tiles, there may be overlap at these
connections. Area 1/4 for the square, 1/3 for the triangle.
e) Partially disconnected tiles, made of infinite "lakes" and "islands".
Area 1/4 for the square, 1/3 for the triangle.
f) Space-filling "dusts", made of disconnected segments. Here, each tile
occupies the reptile fully (area = 1 again; but also = 1/s?!) but still does
not overlap the segments of its 3 fellow tiles except at infinitesimal
points... There are many different space-filling HTs, both connected and
dusts. At the limit they fill in the reptile solid. But careful use of
certain colouring algorithms which I have found for Fractint gives each one
a beautiful distinct inner texture. The most important colouring variation
adds >1 to the first f and <1 to the last one, i.e. f=>1ff+f+f++ff++f+f+<1f.
g) Tiles having all lines parallel - another infinite subset for squares,
but (see a) existing as a unique tile for triangles of any side length. For
squares the area of such tiles is 1/2 instead of the standard 1/4; for the
triangular version it is space-filling, area = 1. These types in squares
also may be "contiguous" or not. They have their own "DNA" (see REPTILES
AND GENETIC ENGINEERING, below).
(I had thought of tiles which are other fractions of a square but related to
the "standard" 1/4, e.g. 1/2 and 1/8. In my ignorance, however, I first
tried to make half-square tiles by using twice the number of segments as for
quarter-square tiles. This was successful only in that my initial shape - a
letter "F" which interlocks with itself upside down - is indeed a possible
tile to programme. But this shape merely filled with ever smaller squares
as the iterations increased - not a satisfactory result at all, having no
fractal edges. So I put aside my search for half-square tiles, not yet
realising that in fact I already had several examples.)
The great surprise for me was to realise that, like square-filling tiles,
half-square tiles need the same number of segments as the standard 1/4
tiles! This is counter to intuition. But the key, the only difference, is
that these new tiles have all lines parallel, either vertical or horizontal.
From such programming an infinite set of half-square tiles develops.
These tiles are useful in that actually only one colour of ink, pixels or
other medium is needed to produce a tiled square: render the tile in, say,
black on a white ground, outline the whole square in black as well, and you
have the pair of tiles - one black, the other white.
h) L-systems allow the use of the symbol "!", which reverses directional
commands. I have found that throwing a "!" into the programming string of
certain square HTs (of side length 2 so far) makes a new tile entirely, one
which has only 1/8 of the area of the square! If paired with its mirror
image, a quarter-square tile results. So far I have not found any examples
of this strange variation in squares above size 2 or in any triangles at
all. Do they exist?
i) There is another kind of eighth-square (sixth-triangle) tile as well,
with a condition usually applying to it. Begin with a standard 1/4 square
(or 1/3 triangle) HT which has reflective symmetry across its vertical axis,
*and* with all its exterior segments on the same straight line. Reverse the
direction of its edge segments; this new tile will be the result. (The
symmetry condition is not , however, required for all-parallel, half-square
types (as in g above). Perform the reversal of the edge segments'
directions here, and the new type of tile (in this case filling a quarter of
a square) will appear. If the original all-parallel tile had no x-axis
reflectional symmetry, then the new tile will have glide symmetry.)
So there we have it: tiles making 1/8, 1/4, 1/2 and 1/1 of a square, and
1/6, 1/3 and 1/1 of a triangle, all using the same proportion of the
reptile's segments! This is amazing.
Several of these subgroups above stretch the idea of "tile" to include
curves made of an infinity of separate pieces, or curves which cross each
other at infinite infinitesimal points. This is unorthodox but, I believe,
possible, allowable; and the beauty of the results also speaks clearly.
Initially I was stuck with using the lower horizontal ("first") side of a
reptile as the set of exterior segments in every case. I had not yet caught
on to the uses of "g", which means "move without drawing". "G" properly
used allows exterior segments to be spread to any position around the
reptile. It also allows for interior segments to be single-sided instead of
the original double-sided ones I used. These two variations showed me that
there is a much greater number of HTs even for a square or triangle of side
length 2 than I had first imagined. I thought that there was only one tile
in each case at this size: how wrong I was.
Very late in the game, I realised that simply rotating the position of a
tile relative to the horizon makes new tiles - although their
first-iteration renderings will be identical. Suddenly, every square tile
had three relatives, every triangle tile two, making rotation sets (RSs), as
I have called them *1. Another burst of programming as I checked out
various details of the new discovery. It seems that the invisible horizon
to which all L-systems silently refer does in fact play a role in their
formation, more than just a visual guideline! There is nothing else to
explain how a tile can look the same at first iteration in three or four
different rotations, yet become completely other tiles as the calculating
continues into higher iterations. Mysterious and wonderful. Some types of
tile are actually duplicated in rotation, either 180 degrees away from
"start" or at every position - these latter types I call "invariant under
rotation".
I also note that rotationally-related HTs may display differences in type
from contiguous to dust to partly 1-sided interiors. (This is due to the
presence of so many extra gs in the formula.) This means that rotations
seem to have their own "dialects": what is beautiful at one angle may be
completely boring or ugly in rotation (see AESTHETICS, below). This would
require that one learn these "dialects" in order to find beautiful HTs in
the various rotations. Also, if an HT rotational set has x-axis
reflectional symmetry in any member, then the 2 members 90 and 270 degrees
away from it will generally be identical except for their rotation, and the
member 180 degrees away will have x-axis reflectional symmetry but be
different in form from the other member with x-axis reflectional symmetry.
The same is for members with 45-degree mirror symmetry: the member 180
degrees away shall have this as well (though it will be different in form);
those 90 and 270 degrees away will be identical except for position. The
only HT so far which remains perfectly invariant (at the limit) under
rotation is the space-filling dust, though this has many different textural
variations. It will be interesting to find any others which have this
property, if they exist. (3/9/2003: NB There is at least one other
non-varying rotation set, starting with the unique (so far) "dust" HT to
have >1 <4 n of pieces (it is a square of size 4 with 2 pieces, each
composed of 2 squares joined diagonally at a corner). Are there other such
pieces with >1 <4 n of pieces?) Other, fractal-edged HTs do come close to
being non-varying in rotation - those with all the action happening along
the square's diagonals. Perhaps there is a "most orthodox" form of these
"diagonal-action" types, which will be called this by virtue of its being
invariant in rotation.)
Rotation sets are an important way of relating different HTs to each other,
as they allow one more easily to come up with a system for finding the total
number of HTs for a particular reptile. The other key is to sytematise
separately all exterior and interior possiblities for the reptile in
question, and then multiply these for a total.
How can I prove an infinity of tiles for the regular reptiles? Well, I'm
actually not enough of a mathematician do do so in a rigorous manner. All I
can say is that side length can be extended from 2 to 3, 4, ... to infinity,
so a single tile in each side length would already lead to an infinite
number of them. But I have found huge numbers of my tiles just in sizes 2-6
- over 6000 of them so far. So the case looks good for infinity.
SOME RULES AND NUMBERS
The number of (standard) Hanmer Tiles required to fill a regular reptile is
equal to s, that reptile's number of sides: e.g. 3 tiles for a triangle (3
sides), 4 for a square (4 sides), 6 for a Cube (6 faces), and thus as a
fraction each such tile (standard types) occupies 1/s of the space of the
reptile.
The number N of segments required to make an HT of a reptile with number of
sides s in d dimensions for iteration number i is calculated: N = (s ^ d) ^
i. With segment totals growing exponentially, vector file sizes can grow
huge; my record is a file over 1 GB in size!
For 2 dimensions:
The total number of exterior segents of a reptile with number of sides s, of
side length L, = Ls.
The ratio of exterior to interior segments = 1 : (s - 1).
The number of segments in a (standard) Hanmer Tile in 1st iteration = s ^ 2.
The total number of elements in the reptile = exterior number + interior
number.
One major calculation which still eludes me is for the total number of HTs
for a given reptile of dimension d, side length L and number of sides s.
(Don't forget, they must use the correct proportion of interior and exterior
segments of the reptile! And not repeat or omit any segment between them!)
Then there is the number of *unique* HTs for the same reptile... With root
side length reptiles' HTs, and various symmetries, cutting down the number
from total to unique, this may turn out to be indeterminable. But I could
be totally wrong in my guess.
REPTILES AND GENETIC ENGINEERING
One of the easy ways to program a large number of HTs in a small time is to
begin with what I call DNA, of which there need be only one sequence for
each reptile and side length. (A-p types have their own modified DNA,
however, to make it easier to program them.) Program an L-system containing
ALL of the segments of the specified reptile in succession, using "g"
instead of "f" for each segment. Then simply copy this master, and "switch
on" the required "genes" (segments) by replacing each "g" with an "f". The
"g"s remain in the DNA sequence, but only as "junk DNA"; all they do is slow
down a calculation without increasing the size of the output file. Fractint
currently (in the latest developer's version) limits the user to a line
length of 225 characters, but this is sufficient for square DNA sequences up
to at least side length 7 and triangle DNA sequences to at least side length
8.
One of the items in my HT zip file at the website listed after this article
is called HTDNA.l. It contains all of my standard "DNA sequences" for HT
squares and triangles up to size 5, to show the progression. (In it, I have
distinguished exterior and interior segments by the use of "G" and "g".)
This is what should become the standard notation of HT DNA in L-systems, for
the purposes of establishing and maintaining uniformity from the start. The
only exceptions to this standard DNA should be a-p types, with their own
modified DNA, and RSs with extra sets of Gs added rather than having to
reprogram the entire HT for each rotation.
-|- -|- -|- -|- -|- -|- -|- -|- -|-
Every reptile whose side length is the exponent of a whole number (which is
its root) will contain in its HTs all of the HTs of reptiles of roots of
that side length. For example, the total set of square HTs of side length 4
(and 8, 16, etc.) includes all HTs of side length 2; size 9 contains all HTs
of size 3; etc. In addition, of course, there will be many tiles in the
exponent size's set which are slight deviations from these root tiles. In
this way, the smaller sizes can be thought of as more fundamental or basic
or primitive than the larger sizes - but not necessarily less beautiful, by
any means. The larger sizes merely offer more and more possibilties to
refine one's tiles.
Which leads me to my next point:
ANIMATIONS
Consider that a square HT of, say, side length 2 and 2nd iteration will be
duplicated in the form of a certain HT of side length 4 - the square of 2 -
at 1st iteration. Again at side length 8, and 16, and so on. Extrapolate
this, and you will have to conclude that between any two HTs of a particular
reptile there will be an infinite number of incremented HTs. These are
found by successively finer and finer adjustments using longer and longer
side lengths. Of course, the programming gets slower and slower as well;
but computers are also gaining speed and memory size, with entirely new
technologies being tested to surpass the looming current physical
limitations.
The result? Animations, as smooth as are required, between any two HTs,
provided that they are of the same "genus", both Tiling the reptile in the
same number of pieces. (This may not work at all for those special tiles
which include "!" in their programming.) A continuous motion, if you would
like it and can program it.
Size, too, is no object. Because we are dealing with fractals which can be
calculated endlessly, if in the future a tile the size of a wall or a
building (yet with detail as small as the eye can resolve) would be desired,
it could be produced.
IFS VERSIONS, IN SHORT
Iterated Function Systems, or IFS, can render HTs just as easily as
L-systems can, and they have their own unique textural and colouring
variations, though the actual programming is best not done in Fractint.
(The exception to what is possible is the "!" types, for which IFS currently
has no equivalent command.) I use Brazil Design version 0.4, which like
Fractint is freeware, but is Windows-based. It can save IFS files for
Fractint use as well as in its own native format. Again, one may begin with
a set of DNA for each reptile and side length. For IFS types, the DNA has
each "gene" segment on a separate line, turned off by default by preceeding
it with a ; semicolon. (I have retained the same segment order as is found
in L-systems HT DNA, in order to minimise confusion and maximise ease of
translating HTs into IFS versions.) Turn on any segment simply by removing
this semicolon. Also, Fractint is a bit limiting here, with a current
maximum of 32 active lines allowed in any IFS. But this still gives squares
up to side length 5. I have not done much with my tiles in IFS versions,
especially in triangles, as the Brazil programme currently does not allow
exact placing of triangles. But the theoretical possibility is there.
TRANSLATIONS
It would be useful to automate the process of re-programming any existing HT
(I have many thousands) as a version of the standard DNA. (But let my
standard L-system drawing methods become the standard way of rendering these
tiles as L-systems, so as to make things regular from the start.) Also,
translations between L-system and IFS versions would be a good idea. Both
of these programming capabilities would not be very difficult to produce,
knowing the rules of HTs, L-systems and IFSs.
ONE DIMENSION...
Of course, linear HTs made of points are also possible. However, they seem
to number exactly one in total for all sizes. This is a dust of points and
the point-size spaces between them, alternating, a variation on Cantor Dust.
This tile has glide symmetry; move it one point to the left or right, and
the spaces and points fill each other. (Yawn)
The point space itself offers no possibilities at all for tiles, of course,
having zero size and dimension...
AESTHETICS; OR, WHAT MAKES AN H-TILE BEAUTIFUL?
With an infinity of H-tiles available to each reptile, there must be a large
number of these which are not worth seeing: pure dusts; duplicates of all
side lengths in those higher side lengths which are exponents of them;
mirror images of other tiles; and just plain ugly tiles. What separates all
of these from those which hold the interest, which have an aesthetic
"value"? For me, at least, there is a minimum of pleasing contiguity and
maximum of aesthetic "dustiness". The tile may, indeed, have 100%
contiguity and still be beautiful. But 0% contiguity equals a perfect
(100%) dust, which is interesting the first time but loses its novelty
fairly fast. Too much scattering of segments also causes a loss of
definition or form, leading to fuzziness. There should be some areas of the
tile which are solid, or it literally will have no shape except that of the
entire reptile itself.
So there is good news and bad news. The bad news is, there's an infinity of
rubbish tiles among the endless HTs of any reptile. The good news is,
there's also an unending collection of beautiful, sublime and wonderful
tiles also awaiting discovery. Only patient exploration can root out the
latter from the former. Because we have specified that these must all be
tiles of reptiles, however, our infinite sets will contain none of the Mona
Lisas or other similar masterpieces of human art found in the "larger"
infinity of a square of entirely unlimited interior possibilities. This is
precisely because of the fact that only these tiles to which we have limited
our search are in our sets. Infinity does not have to mean "all things".
(There may, however, be some close approximations to the tiling works of
M.C. Escher: one of my favourite artists, no surprise there. This
Multiverse of tiles may, indeed, be thought of as an idealisation of
Escher's wonderful tilings. There are no representations of actual
creatures in these tiles, so in this way they are different from his work,
which mapped the mathematical onto a world of representational things or
objects. And this lack means that Hanmer Tiles are less pictorial and are
purely mathematical. They have no anchor in the real world, one might say.
Escher's genuis was, I believe, precisely in the mix of the mathematical and
the pictorial, something he consciously strove for.)
Hanmer Tiles, and fractals in general, are definitely images for the machine
age. They begin with mathematical fomulae; they can best be rendered by the
brute force of number-crunching; and their typical intricacies from
exponential growth are a nightmare for any kind of hand calculation or
rendering process, whether it be carpet weaving, mosaic laying, Lego or
whatever. Does this alone make them not art? I think not. Even if they
are not destined to be included in one's definition of art, however, this
does not mean that one is forbidden from calling them beautiful.
Mathematical beauty may be accused of having a purity which could be called
inhuman. Perhaps the eye longs for a break in the pattern somewhere, rather
than sterile perfection! No real tree is an exact L-system, after all, and
no L-system (as far as I know) looks as real as a tree. But mathematical
beauty perhaps calls to the part of us which longs for the perfect, and
frustrates the part which is satisfied with less.
What to do with these tiles? I can't do it all myself, that's for certain!
Push on deeper into the universe which each one contains in the search for
more undiscovered territory. (Think of an analogy with the Mandelbrot Set:
ever larger reptile sizes correspond to deeper zooms, taking more time to
calculate.) Programme a computer search by calculating automatically,
possibly using the HT DNA - the rules are simple enough. Have the total of
*unique* tiles collected into a separate file. But *people* will still have
to choose to concentrate on those finds which are more pleasing to the eye
and pass over the rest. Use them wherever Tiling designs are wanted. Allow
a subtle variation in a large set of similar tiles instead of keeping them
all the same from one edge of the set to the other, sort of a parquet
deformation.
Three dimensions? I long for the day when I will hold in my hand a clear
cube, say 30 cm on a side, containing 6 identical tiles, each of these
differing only in its transparent colour. The whole thing thus will be
visible all the way through. Its code will have been programmed in I don't
yet know which programme. I am not content with seeing the thing rotating
on some 2-dimensional screen; it will have been "printed" by an extension of
our current 2-d printing processes, and will be a solid cube.
The next step will be an animated version of the same thing, its 3-d "tiles"
set in motion by a fluid of nano-particles inside the cube, a fluid in 6
colours which can instantly change its state to become solid when instructed
to do so. It will display smooth transitions from one tile to the next at
whatever speed the user desires. Or, alternately, it might be made of a
stack of 2-d square HTs, each slightly differing from thise above and below
it. These could cycle from bottom to top and around again to make the
animation.
The size 2 cube, then. Composed of 2 x 2 x 2 = 8 cubes, each having 6
square faces for a total of 48 faces (the analogy of our 2-d segments), and
each having 12 edges for a total of 96, and 8 vertices for a total of 64.
Each smaller cube meets 3 others at various faces, a different 3 others at
edges, and all 7 others at a single vertex. Each side of the whole cube has
(is divided into) 4 faces, so 4 x 6 = 24 faces - half of the total for size
2 - are exterior, the other 24 interior. Each HT here will have 48 / 6 = 8
face "segments" - 4 exterior and 4 interior.
Have we long to wait?
Anthony Hanmer
Tbilisi, Republic of Georgia
*1 I chose to label the members of these rotation sets as follows: the
orginal HT would have no suffix, just its usual number; The RS members
would be suffixed a, b, c for squares and a, b for triangles; for cubes they
would thus be a, b, c, d, e. Other suffixes I have decided on include ~ for
"!" variations and other oddities; * for contiguous HTs; # for dusts of any
kind except space-filling dusts; ^ for space-filling dusts; k, l, m and n
for colouring variations; r for e-r (edge-reversed) variations. I ended up
losing the */# distinction in the end, through sheer carelessness as the
number of my tiles rapidly increased. But it was originally my intention.
PS The L-system and IFS files associated with this article can be found
(zipped as HanmerTiles.zip) at
http://spanky.triumf.ca/pub/fractals/LSYSTEMS/
They are named as follows:
HTSq23.l
HTSq4a.l
HTSq4b.l
HTSq5.l
HTSq6a.l
HTSq6-a.l
HTSq6-b.l
HTSq6-c.l
HTSq_a-p.l
HTTr.l
HTDNA.l
HTs.ifs
HTDNA.ifs
Download and unzip the files into your Fractint directory, and they're ready
for use as either type Lsystem or IFS.
Part II of this posting is some examples as L-systems.
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3
I presume you've seen Part II by now - some of my best finds so far. Not
par files, just L-system programming.
Tony Hanmer
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EXAMPLES
These are taken from my collection of over 6000 Hanmer Tiles, to show their
diversity. Paste the text below into a text editor and call it "HT_Eg.l",
saving it in your Fractint folder. Then you're ready to try the pieces in
Fractint as type Lsystem. (NB If any of the longer lines of these examples
wrap onto 2 or more lines, they must end up as 1 line again to work in
Fractint.)
sq20004* { ; Anthony Hanmer 11/10/2002
Angle 4 ; 4th 2x2 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f; Unexpected (as 1-sided interior) but true
f=f+f+f|g-g+f
g=gg
}
sq20005~ { ; Anthony Hanmer 15/10/2002
Angle 4 ; 5th 2x2 square tile...
Axiom f ; when put with its mirror image.
; One of my VERY best HTs
; X-axis reflectional symmetry
f=f+!f|!f+f
}
sq20005r { ; Anthony Hanmer 9/9/2003
Angle 4 ; 5th 2x2 square tile...
Axiom f ; when put with its mirror image.
; One of my VERY best HTs
; X-axis reflectional symmetry
f=g|f|g+!f|!f+g|f|g
g=gg
}
sq20006~ { ; Anthony Hanmer 15/10/2002
Angle 4 ; 6th 2x2 square tile...
Axiom f ; when put with its mirror image.
; X-axis reflectional symmetry
; One of my VERY best HTs
f=f+!gf|!fg+f
g=gg
}
sq20006b { ; Anthony Hanmer 4/9/2003
Angle 4 ; 180-degree rotation
Axiom f ; This IS a new 1/2-Sq flip tile.
; X-axis reflectional symmetry
f=gg+gg+f+!gf|!fg+f+gg+gg
g=gg
}
sq20014# { ; Anthony Hanmer 19/10/2002
Angle 4 ; 14th 2x2 square HTile, 8th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff+g+gf+gg
g=gg
}
sq20021* { ; Anthony Hanmer 19/10/2002
Angle 4 ; 21st 2x2 square HTile
; (1b)
; One of my VERY best HTs
; X-axis reflectional symmetry
Axiom c53f+c2f+c12f+c14f
f=gg+gg+f+f|f+f+gg+gg
g=gg
}
sq20023* { ; Anthony Hanmer 19/10/2002
Angle 4 ; 23rd 2x2 square HTile
; (4b)
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=gg+gg+f+f+f|g-g+f+gg+gg
g=gg
}
sq20033* { ; Anthony Hanmer 22/10/2002
Angle 4 ; 33rd 2x2 square HTile, a-p
; (2b)
; One of my VERY best HTs
Axiom c53f+g+c12f
f=gg+g+f|f+g+ff+gg+gg
; f=g+g+f|f-g+g+gg+ff+gg+gg; formerly
g=gg
}
sq20034# { ; Anthony Hanmer 22/10/2002
Angle 4 ; 34th 2x2 square HTile, 17th dust
; (9b)
; One of my VERY best HTs:
; One of the most picturesque square HTiles of side 2
Axiom c53f+c2f+c12f+c14f
f=gg+g+f|f+g+fg+gf+gg
; f=g+g+f|f-g+g+fg+gf+gg+gg; formerly
g=gg
}
sq20052# { ; Anthony Hanmer 23/10/2002
Angle 4 ; 52nd 2x2 square HTile, 33rd dust
; (12b)
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=g+f+f+f+gg+gg+fg+gg+gg
g=gg
}
sq20057# { ; Anthony Hanmer 23/10/2002
Angle 4 ; 57th 2x2 square HTile, 36th dust
; ()
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+f-f|g+g+g+gg+gg+gf+gg
g=gg
}
sq20077~ { ; Anthony Hanmer 9/9/2003
Angle 4 ; 2x2 square tile (almost)...
Axiom f ; when put with its mirror image.
; One of my interesting HTs
f=f+!gf|!gf+f
}
sq20078~ { ; Anthony Hanmer 9/9/2003
Angle 4 ; 2x2 square tile...
Axiom f ; when put with its mirror image.
; (45-degree reflection needed)
; One of my VERY best HTs
f=f+!ff|!gg+f
g=gg
}
sq20078b { ; Anthony Hanmer 4/9/2003
Angle 4 ; 180-degree rotation
Axiom f ; 1/2 a square, tiles with
; its 45-degree reflection
; One of my VERY best HTs
f=gg+gg+f+!ff|!gg+f+gg+gg
g=gg
}
sq30001* { ; Anthony Hanmer 2000
Angle 4 ; My very first HT
; (& 1st 3x3 square HT)
; Before I knew the rules of HTs...
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=ff+f+f++ff++f+f+f
}
sq30001c { ; Anthony Hanmer 17/8/2003, 15:00
Angle 4 ; Whoa, this is BIG:
; Merely rotating each HT gives other new HTs!!!
; One of my best HTs: 270 degrees
Axiom c53f+c2f+c12f+c14f
f=ggg+ggg+ggg+ff+f+f++ff++f+f+f+ggg
g=ggg
}
sq30002* { ; Anthony Hanmer 2000
Angle 4 ; My 3rd HT
; (2nd 3x3 square EDr)
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+f+f++f+f++ff+ff
}
sq30003* { ; Anthony Hanmer 2000
Angle 4 ; My 4th HT
; (3rd 3x3 square HT)
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff++ff+f+f++f+f
}
sq30003a { ; Anthony Hanmer 17/8/2003
Angle 4 ; 90-degree rotation of my 3rd HT
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=ggg+f+ff++ff+f+f++f+f+ggg+ggg+ggg
g=ggg
}
sq30006# { ; Anthony Hanmer 5/10/2002
Angle 4 ; 6th 3x3 square HTile, 1st dust
; One of my VERY best HTs
Axiom c53f+g+c12f
; (25/7/2003: This axiom proves that, for HTs with only
; parallel lines, only 2 tiles are needed, not 4)
f=f+g+f|fgf|fg-g-f|f+gg+ff
g=ggg
}
sq30029# { ; Anthony Hanmer 6/10/2002
Angle 4 ; 29th 3x3 square HTile, 24th dust
; One of my VERY best HTs
; X-axis reflectional symmetry
Axiom c53f+g+c12f
f=f+g-f|f-g+f|fgf|fg+gg+ff
g=ggg
}
sq30453# { ; Anthony Hanmer 14/8/2003
Angle 4 ; 3x3 square HTile, a-p dust, tiles in only 2
; One of my VERY best HTs: WAY cool
Axiom c53f+g+c12f
f=g+g+f|f+g+f|f-g+f|f+g+ff+ggg+fgg+ggg+ggg
g=ggg
}
sq30454# { ; Anthony Hanmer 14/8/2003
Angle 4 ; 3x3 square HTile, a-p dust, tiles in only 2
; One of my VERY best HTs: WAY cool
Axiom c53f+g+c12f
f=f+g+f|f+g+f|f-g+f|f+g+gf+ggg+gfg+ggg+ggg
g=ggg
}
sq30475# { ; Anthony Hanmer 14/8/2003
Angle 4 ; 3x3 square HTile, a-p dust, tiles in only 2
; One of my VERY best HTs: WAY cool
Axiom c53f+g+c12f
f=fgf+g+fgg|ffg+gg+gfg+g+gff|ggf+gg+ggg
g=ggg
}
sq30558# { ; Anthony Hanmer 17/8/2003
Angle 4 ; 3x3 square HTile, a-p dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq3 DNA
; X-axis reflectional symmetry
Axiom c53f+g+c12f
f=FFF+G+fgf|fff+GG+GGG+G+gfg|ggg+GG+GGG
g=ggg
}
sq30564# { ; Anthony Hanmer 17/8/2003
Angle 4 ; 3x3 square HTile, a-p dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq3 DNA
Axiom c53f+g+c12f
f=FFF+G+ggf|fgf+GG+GGG+G+ffg|gfg+GG+GGG
g=ggg
}
sq30581# { ; Anthony Hanmer 17/8/2003
Angle 4 ; 3x3 square HTile, a-p dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq3 DNA: variation on 30557
Axiom c53f+g+c12f
f=GFG+G+gff|fff+GG+FGF+G+fgg|ggg+GG+GGG
g=ggg
}
sq30582# { ; Anthony Hanmer 17/8/2003
Angle 4 ; 3x3 square HTile, a-p dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq3 DNA: variation on 30557
Axiom c53f+g+c12f
f=FGG+G+gff|fff+GG+GFF+G+fgg|ggg+GG+GGG
g=ggg
}
sq40001* { ; Anthony Hanmer 10/9/2002
Angle 4 ; And now, the 1st 4x4 square HTile!
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+f+f|f+f-f|f+ff+f+f+f|f-f+ff
}
sq40003* { ; Anthony Hanmer 10/9/2002
Angle 4 ; 3rd 4x4 square HTile!
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f; (See 8914)
f=f+fff|f+f|ff|f-f-f|f-f+fff
}
sq40005* { ; Anthony Hanmer 10/9/2002
Angle 4 ; 5th 4x4 square HTile!
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+f-f+f|f-f-f|f-f|f-f+f+f|f+ff
}
sq40006* { ; Anthony Hanmer 10/9/2002
Angle 4 ; 6th 4x4 square HTile!
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+fff|ff-f|f-f+f+ff|ff+ff
}
sq40011* { ; Anthony Hanmer 10/9/2002
Angle 4 ; 11th 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+fff|f+f|f+f-f|f-f+f+f|f+ff
}
sq40013* { ; Anthony Hanmer 12/9/2002
Angle 4 ; 13th 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff+f|f-f-f|ff+f|f-f+f+fff
}
sq40018* { ; Anthony Hanmer 18/9/2002
Angle 4 ; 18th 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff+f|f-ff+ff+ff+f|f-ff+f
}
sq40019* { ; Anthony Hanmer 18/9/2002
Angle 4 ; 19th 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff-f|f+ff+f+f|f+f+ff|ff+f
}
sq40020* { ; Anthony Hanmer 18/9/2002
Angle 4 ; 20th 4x4 square HTile
; One of my VERY best HTs
; Seems to contain a 180-degree symmetrical half,
; and 2 quarters which make this same shape together...
Axiom c53f+c2f+c12f+c14f
f=f+fff|f-f|ff|f+ff+ff+f|f+f
}
sq40021* { ; Anthony Hanmer 19/9/2002
Angle 4 ; 21st 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=ff+f+f-f-f|f+f+f-f+f+f-f|f+f+f
}
sq40021c { ; Anthony Hanmer 25/8/2003
Angle 4 ; 270-degree rotation
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=gggg+gggg+gggg+ff+f+f-f-f|f+f+f-f+f+f-f|f+f+f+gggg
g=gggg
}
sq40022* { ; Anthony Hanmer 19/9/2002
Angle 4 ; 22nd 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=ff+f+f-f|f+f+f|ff+f+f-f|f+f+f
}
sq40023* { ; Anthony Hanmer 19/9/2002
Angle 4 ; 23rd 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+fff|f+f|f+ff+f+f|f+f+f|f+f
}
sq40024* { ; Anthony Hanmer 19/9/2002
Angle 4 ; 24th 4x4 square HTile
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff+f|ff|f+f+f|f+f+ff+f|f+f
}
sq40025# { ; Anthony Hanmer 5/10/2002
Angle 4 ; 25th 4x4 square HTile, 1st dust
Axiom c53f+c2f+c12f+c14f
; One of my VERY best HTs
f=f+fg+f|ff|f+g-f|fff|ff+f+fff
g=gggg; THIS allows "tile dusts"!
; (N of gs equalling g is n of side units
; in 1st iteration)
}
sq40026# { ; Anthony Hanmer 7/10/2002
Angle 4 ; 26th 4x4 square HTile, 2nd dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+fgf-f-f|f+f+fgf+f+f|f+f+f|f+f
g=gggg
}
sq40031# { ; Anthony Hanmer 7/10/2002
Angle 4 ; 31st 4x4 square HTile, 7th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+f|f+f+ff|ff+f+fgf+fg|gf-fgf+f
g=gggg
}
sq40036# { ; Anthony Hanmer 7/10/2002
Angle 4 ; 36th 4x4 square HTile, 12th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+f|f+f+fg+f|f-gf+f+f+f|f+gg+fg|gf-ggf+f
g=gggg
}
sq40062# { ; Anthony Hanmer 8/10/2002
Angle 4 ; 62nd 4x4 square HTile, 38th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+gggf|f-f|f-ggg+f+f+f-f-f|f+f+f-f+ff
g=gggg
}
sq40064# { ; Anthony Hanmer 8/10/2002
Angle 4 ; 64th 4x4 square HTile, 40th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ggg+f|fggf|fgg+ggg+f+f+f-f-f|f+f+f-f+ff
g=gggg
}
sq40066# { ; Anthony Hanmer 8/10/2002
Angle 4 ; 66th 4x4 square HTile, 42nd dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ggg+f|f-ggg+f+f+f-f-f|f+f+f-f+f+f|f+f
g=gggg
}
sq40070# { ; Anthony Hanmer 8/10/2002
Angle 4 ; 70th 4x4 square HTile, 46th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+fggf|fggf+f+f+f-f-f|f+f+f-f+ff
g=gggg
}
sq40071# { ; Anthony Hanmer 8/10/2002
Angle 4 ; 71st 4x4 square HTile, 47th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+f|f+f+f+f-f-f|f+f+f-f+f+ggg-f|f+ggg+f
g=gggg
}
sq40081# { ; Anthony Hanmer 8/10/2002
Angle 4 ; 81st 4x4 square HTile, 57th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+gff|f+f|f+f-f|f-g+f+f|f+f+g-f|f+g+f
g=gggg
}
sq40089# { ; Anthony Hanmer 9/10/2002
Angle 4 ; 89th 4x4 square HTile, 65th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff-gf|fg+f-f|ff|f+f+f+f|f+ff
g=gggg
}
sq40090# { ; Anthony Hanmer 9/10/2002
Angle 4 ; 90th 4x4 square HTile, 66th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+g-ff|ff+g+f+fgf|fgf+f+f-f|f+f+f
g=gggg
}
sq40092# { ; Anthony Hanmer 10/10/2002
Angle 4 ; 92nd 4x4 square HTile, 68th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+g-ff|ff+g+f+fg+f|f-gf+f+f-f|f+f+f
g=gggg
}
sq40093# { ; Anthony Hanmer 10/10/2002
Angle 4 ; 93rd 4x4 square HTile, 69th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+g-ff|ff+g+f+fg-f|f+gf+f+f-f|f+f+f
g=gggg
}
sq40115# { ; Anthony Hanmer 9/10/2002
Angle 4 ; 115th 4x4 square HTile, 91st dust
; One of my VERY best HTs
; X-axis reflectional symmetry
Axiom c53f+c2f+c12f+c14f
f=f+g-ff|ff+g+f+fgf|fgf+f+ggg-f|fggf|fgg-ggg+f
g=gggg
}
sq40132# { ; Anthony Hanmer 17/10/2002
Angle 4 ; 132nd 4x4 square HTile, 108th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff|f-f|ff|f+f+f+fgf|fgf+ff
g=gggg
}
sq40182# { ; Anthony Hanmer 17/10/2002
Angle 4 ; 182nd 4x4 square HTile, 159th dust
; One of my VERY best HTs
; X-axis reflectional symmetry
Axiom c53f+c2f+c12f+c14f
f=ff+f+f|f+f|f+f|f+f+f+ggg+ggf|fggf|f+ggg+f
g=gggg
}
sq40198# { ; Anthony Hanmer 18/10/2002
Angle 4 ; 198th 4x4 square HTile, 175th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+ff-gf|fg+f-f|f-f+f+f|f+f+gf|fg+f
g=gggg
}
sq40220# { ; Anthony Hanmer 18/10/2002
Angle 4 ; 220th 4x4 square HTile, 196th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+gg-f|f+gg+f+f+f|f-f+f+fgf|fg+f|f+f+f
g=gggg
}
sq40221# { ; Anthony Hanmer 18/10/2002
Angle 4 ; 221st 4x4 square HTile, 197th dust
; One of my VERY best HTs
Axiom c53f+c2f+c12f+c14f
f=f+gg-f|f+gg+f+f+f|f-f+f+fgg+gf|fg-gg+f|f+f+f
g=gggg
}
sq41262# { ; Anthony Hanmer 15/8/2003
Angle 4 ; 4x4 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq4 DNA
Axiom c53f+g+c12f
f=FFFF+G+fffg|ggff+G+fg|gf+GG+GGGG+G+gggf|ffgg+G+gf|fg+GG+GGGG
g=gggg
}
sq41265# { ; Anthony Hanmer 15/8/2003
Angle 4 ; 4x4 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq4 DNA
Axiom c53f+g+c12f
f=FFFF+G+fffg|ggff+G+gf|gf+GG+GGGG+G+gggf|ffgg+G+fg|fg+GG+GGGG
g=gggg
}
sq41320# { ; Anthony Hanmer 16/8/2003
Angle 4 ; 4x4 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq4 DNA: variation on 41135
Axiom c53f+g+c12f
f=FGFF+G+fgfg|gfgf+G+gg|ff+GG+GFGG+G+gfgf|fgfg+G+ff|gg+GG+GGGG
g=gggg
}
sq41559# { ; Anthony Hanmer 16/8/2003
Angle 4 ; 4x4 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq4 DNA: variation on 41151
Axiom c53f+g+c12f
f=GFFF+G+ffgg|ggff+G+gg|ff+GG+FGGG+G+ggff|ffgg+G+ff|gg+GG+GGGG
g=gggg
}
sq41566# { ; Anthony Hanmer 16/8/2003
Angle 4 ; 4x4 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq4 DNA: variation on 41151
Axiom c53f+g+c12f
f=GFGF+G+ffgg|ggff+G+gg|ff+GG+FGFG+G+ggff|ffgg+G+ff|gg+GG+GGGG
g=gggg
}
sq41574# { ; Anthony Hanmer 16/8/2003
Angle 4 ; 4x4 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq4 DNA: variation on 41152
Axiom c53f+g+c12f
f=GFFF+G+ffgg|ggff+G+fg|gf+GG+FGGG+G+ggff|ffgg+G+gf|fg+GG+GGGG
g=gggg
}
sq41583# { ; Anthony Hanmer 16/8/2003
Angle 4 ; 4x4 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best HTs, using a modified Sq4 DNA: variation on 41152
Axiom c53f+g+c12f
f=GFFG+G+ffgg|ggff+G+fg|gf+GG+FGGF+G+ggff|ffgg+G+gf|fg+GG+GGGG
g=gggg
}
sq50172# { ; Anthony Hanmer 23/11/2002
Angle 4 ; 172nd 5x5 square HTile, dust
; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=f+gf|fg+f+fff|f-f|f-f-f|f-f+f+f+f|f-f+f+f|f+f+ggggg+ggggg+g+f|f+gggg+ggggg
g=ggggg
}
sq50173# { ; Anthony Hanmer 23/11/2002
Angle 4 ; 173rd 5x5 square HTile, dust
; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=f+ff|ff+f+gff|f-f|f-f-f|f-g+f+f+f|f-f+f+f|f+f+gg+f|f+ggg+ggggg+ggggg+ggggg
g=ggggg
}
sq50174# { ; Anthony Hanmer 23/11/2002
Angle 4 ; 174th 5x5 square HTile, dust
; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=f+ff|ff+f+gff|f-f|f-f-f|f-g+f+f+f|f-f+f+f|f+f+ggggg+gg+f|f+ggg+ggggg+ggggg
g=ggggg
}
sq50175# { ; Anthony Hanmer 23/11/2002
Angle 4 ; 175th 5x5 square HTile, dust
; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=f+ff|ff+f+gff|f-f|f-f-f|f-g+f+f+f|f-f+f+f|f+f+ggggg+ggggg+gg+f|f+ggg+ggggg
g=ggggg
}
sq50217# { ; Anthony Hanmer 3/8/2003
Angle 4 ; 5x5 square HTile, all-lines-parallel dust, tiles in only 2
; One of my VERY best Sq5 HTs of this type: WAY COOL, a personal a-p
favourite
Axiom c53f+g+c12f
f=FFFFF+G+ffggf|fggff+G+fgfgg|ggfgf+G+ffgfg|gfgff+G+gffgg|ggffg+G+GGGGG+GGGGG+GGGGG
g=ggggg
}
Sq60727 { ; Anthony Hanmer 24/5/2004
Angle 4 ; Sq6 HT; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=F+fff|fff+F+ffff|ffff+F+f|f+F+ff|ff+F+f|f+F+GGGGGG+GGGGGG+GGG+gfgf|fgfg+GG+gff|ffg+G+GGGGGG
g=gggggg
}
Sq60728 { ; Anthony Hanmer 24/5/2004
Angle 4 ; Sq6 HT; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=F+fff|fff+F+ffff|ffff+F+f|f+F+ff|ff+F+f|f+G+GGGGGG+GGGGGG+GGG+gfgf|fgfg+GG+gff|ffg+F+GGGGGG
g=gggggg
}
Sq60729 { ; Anthony Hanmer 24/5/2004
Angle 4 ; Sq6 HT; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=F+fff|fff+F+ffff|ffff+F+f|f+F+ff|ff+G+f|f+G+GGGGGG+GGGGGG+GGG+gfgf|fgfg+GF+gff|ffg+F+GGGGGG
g=gggggg
}
Sq60839 { ; Anthony Hanmer 24/5/2004
Angle 4 ; Sq6 HT; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=F+fff|fff+F+fffg-gf|fg+gfff+F+f|f+F+gg|ff+F+f|f+F+GGGGGG+GGGGGG+GGG+gff|ffg+G+ff|gg+G+gff|ffg+G+GGGGGG
g=gggggg
}
Sq60840 { ; Anthony Hanmer 24/5/2004
Angle 4 ; Sq6 HT; One of my VERY best HTs: WAY cool
Axiom c53f+c2f+c12f+c14f
f=F+fff|fff+F+fffg-gf|fg+gfff+F+f|f+F+gg|ff+F+f|f+G+GGGGGG+GGGGGG+GGG+gff|ffg+G+ff|gg+G+gff|ffg+F+GGGGGG
g=gggggg
}
tr20001* { ; Anthony Hanmer 2000
Angle 6 ; My 2nd HT
; (& 1st 2x2 triangular HT)
; I think there are no more 2x2 triangular HTiles to be found.
; (I am later proved to be SO wrong.)
; One of my VERY best HTs
Axiom c9f++c12f++c14f
f=f+f|f++f
}
tr20003* { ; Anthony Hanmer 11/10/2002
Angle 6 ; 3rd 2x2 triangular HTile
; 1-sided interior
; One of my VERY best HTs
Axiom c9f++c12f++c14f; Unexpected but true
f=f+f|g--f|g+f
g=gg
}
tr20009* { ; Anthony Hanmer 19/10/2002
Angle 6 ; 9th 2x2 triangular HTile
; One of my VERY best HTs: sheer simplicity
Axiom c9f++c12f++c14f
f=f+f|g--f++f++gg
g=gg
}
tr20014* { ; Anthony Hanmer 23/10/2002
Angle 6 ; 14th 2x2 triangular HTile
; X-axis reflectional symmetry
; One of my VERY best HTs: again, very simple
Axiom c9f++c12f++c14f
f=gg++g+f|f++f++fg++gg
g=gg
}
tr30001* { ; Anthony Hanmer 9/9/2002
Angle 6 ; Eureka! The 2nd triangular HT!
Axiom c9f++c12f++c14f; (1st 3x3 triangular HT)
; One of my VERY best HTs
f=f++f|f++ff|ff++ff
}
tr30002* { ; Anthony Hanmer 9/9/2002
Angle 6 ; Eureka! The 3rd triangular HT!
Axiom c9f++c12f++c14f; (2nd 3x3 triangular HT)
; One of my VERY best HTs
f=f++f--f--f|f++f++f+ff
}
tr30009# { ; Anthony Hanmer 5/10/2002
Angle 6 ; 9th 3x3 triangular HTile, 1st dust
Axiom c9f++c12f++c14f; WOW - "Koch (variation) Dust Tile"...!
; One of my VERY best HTs
; X-axis reflectional symmetry
f=f+fg++f|f--gf++f++f|f+f
g=ggg
}
tr30010# { ; Anthony Hanmer 5/10/2002
Angle 6 ; 10th 3x3 triangular HTile, 2nd dust
Axiom c9f++c12f++c14f; WOW - "Sierpinski (variation) Dust Tile"...!
; One of my VERY best HTs
; X-axis symmetry
f=f+gf++f++f|f--f--fg++ff
g=ggg
}
tr30221* { ; Anthony Hanmer 24/5/2003
Angle 6 ; 221st 3x3 triangular HTile
; 1-sided interior
; One of my VERY best HTs
Axiom c9f++c12f++c14f
f=f+f++f|g--g++f+f|g--f--f--g++f++ggg++gg+g|f++g++ggg
g=ggg
}
tr30244* { ; Anthony Hanmer 24/5/2003
Angle 6 ; 244th 3x3 triangular HTile
; 1-sided interior
; One of my VERY best HTs
Axiom c9f++c12f++c14f
f=f++f|g++f--f|g++g++fg++g++g|f++g--g|f++f++gg++ggf++ggg
g=ggg
}
tr40071* { ; Anthony Hanmer 8/11/2002
Angle 6 ; 71st 4x4 triangular HTile
; One of my VERY best HTs
Axiom c9f++c12f++c14f
f=f++f--f--f|f++f++f+f++f--f--f|f++f++f+fg++gggg++gggf++gggg
g=gggg
}
tr40071a { ; Anthony Hanmer 13/9/2003
Angle 6 ; 120-degree rotation
; One of my VERY best HTs
Axiom c9f++c12f++c14f
f=gggg++f++f--f--f|f++f++f+f++f--f--f|f++f++f+fg++gggg++gggf++gggg++gggg++gggg
g=gggg
}
tr40398# { ; Anthony Hanmer 15/9/2003
Angle 6 ; 4x4 triangular HTile
; Variation on Tr40233, re-drawn using the L4 DNA
; One of my VERY best HTs
Axiom c9f++c12f++c14f
f=F+ffg|ggf++G+fg|gf++F+g|g++G++G+ggf|gfg++G+gg|gg++G+g|g++G++G+ggg|fgg++F+gf|fg++G+f|f++F++GGGG
g=gggg
}
tr40400# { ; Anthony Hanmer 15/9/2003
Angle 6 ; 4x4 triangular HTile
; Variation on Tr40233, re-drawn using the L4 DNA
; One of my VERY best HTs
Axiom c9f++c12f++c14f
f=F+ggg|ggg++F+fg|gf++G+g|g++G++G+fff|gff++G+gg|gg++G+g|g++G++G+ggg|fgg++G+gf|fg++F+f|f++F++GGGG
g=gggg
}
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Test, as my last 2 attempts at a posting failed, maybe because it was too
large.
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FOTD -- September 01, 2004 (Rating 5)
Fractal visionaries and enthusiasts:
I am currently involved in unpacking and setting up a new
graphic-design computer for Fractal Central, so the discussion
for today will be terse. Nothing unusual in that.
For the first FOTD of September I wanted a nice fractal, one
that would be interesting. I wanted something that could be
found in a hurry. It did not need to render in under a minute,
as I could always do other things while the image was rendering.
But above all else, it needed to be nice. I found such an image
in the fractal created by subtracting (Z) from 2 parts of Z^(-2)
and adding (1/C).
Today's image is nice. I leave it to the viewer to decide what
the word 'nice' means when applied to a fractal. And the image
is also interesting. 'Interesting' is what one says about some-
thing when they do not really like it but do not want to offend
its creator.
I named the image "A Nice Fractal" because it is a nice fractal.
I do kind of like the sawtooth effect surrounding the unusually
prominent midget. I rated it at a 5 because it is a nice
fractal. But when the render time of over 42 minutes is consi-
dered, the overall value rates a measly 12, which is not so
nice.
But overall values are subjective things at best, and today's
rather low value may be bypassed by downloading the finished
image from:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
Here at Fractal Central, Monday was cloudy, but the rain stayed
away. Tuesday was partly cloudy with a temperature of 84F 29C.
The cats enjoyed ample time in the yard on both days. Today is
starting sunny and pleasant. I expect a repeat. For me, things
will not settle down until I get the new machine loaded and
running. It is yet to be determined how well it will work with
Fractint. Until next FOTD, take care, and fractals are not real
but they do exist.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
A_Nice_Fractal { ; time=0:42:38.21--SF5 on a P200
reset=2003 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix2 function=recip passes=1
center-mag=+0.21468133873538990/-0.284332453475336\
20/7004249/1/-65/1.09182119595319094e-009
params=-1/1/2/-2/0/0/0/0 float=y
maxiter=7200 inside=0 periodicity=10
colors=000`FSbEQdDOfCMhBKjAIl9Gn8Em7Co7Aq78q70r77r\
7Dr7Js7Ps7Ws7at7gt7mt7sr9qqApoBonCmlDlkEkiFihHhfIg\
eJecKdbLc`Ma_N`YP_XQZVRXUSWSTVRUTPVSOXRMYPLZOJ_NI`\
LGaKFbJAeKEcIIaHM_GPYFTXEXVC_TBcRAgP9iM9jO8jP8jR8j\
S8jU7jV7jX7jY7j_7j`6jb6jc6je6jf5jh5ji5jk5jl5jn4jo4\
jq4jr4jt3ju3jw3jx3jy3hw6gv8etBdsDbrFapI_oKZmNXlPWk\
RUiUThWRfZQe`OdbNbeLagK_jIZlHYnMYlQYjUYhYYgaYefYcj\
YbnY`rYZvYYlZccZhW`oVZmVYlVXjVWiUVgUUfUSeURcTQbTP`\
TO_TNYTMXSKWSJUSITSHRRGQRFORDNRCMQBKQAJQ9HQ8GTA7Q7\
FO4NM1VR2SW2Q`2Od3Mi3Kn3Iq1Dr3Gr5Ir7Ls9NsBQsDStFVt\
HXtJ_tLauNduPfuRivTkvVnvXpvYrtZor_lp_jn`gl`ejabha`\
fbYdcWbcTadR_dOYeMWeJUfHSgEQgCOh9Mh7Ki4Kj0Ji2Ji4Jh\
6Jh8JgAJgCJfEJfFJeHJeJJdLIdNIcPIcRIbTIbUIaZIaYI`aI\
`aI`dI_eI`fJ`gJ`gK`hK`hK`hL`iL`iL`iL`jL`jL`jL`kKak\
K`lKalK`lK`mK_mKZmKYnKXnKWnJVnKUmLTlLSkMRjNQiNPhOO\
gPNfNMePLcRKaTJ_VIYXHWZGU }
frm:MandelbrotMix2 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)),
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l }
END 20.0 PAR-FORMULA FILE==================================
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