Falling somewhere between ambigrams, autological words, concrete poetry and graphic design, but with a definite mathematical
slant, these seem to be a new artform, albeit (at least the ones I do) requiring varying amounts of math background to really appreciate. Some of these,
particularly those in autologlyphs1.jpg, are more like puzzles - to work out what the words say, and are probably impossible to 'get'
without knowing about the concepts described. The name, 'autologlyph' (previously 'mathfont') is with reference to autological words, and
they could equally be called 'autological images'.
On many of these, features at the level of individual pixels are important, so the scaled down thumbnails don't really do all of
them justice.
(January 2002) The first few. Strangely, the hardest 'puzzle' to work out was the first autologlyph I thought of.
(January 2002) These are easier to read, some neat ideas.
(January 2002) Originally this had an extra level (so it was 3 times as big) and was uniform across all
letters. This works better however - it shows the process, and gets out of the problem of trying to depict
a nowhere dense set using finitely many sets with nonempty interior (the pixels).
(January 2002) Another process, this time with a space-filling curve. Yes, it did take forever to do...
(January 2002) The 'C' used to be two self-similar spirals, and although that worked with the mathematical theme, it turned
out too smooth-looking in relation to the other letters. The 'R' was also difficult. The full size picture
is too large to print out on one sheet, so here are the left half,
and the right half.
(January 2002) Top is another puzzle, illustrations originally in "..... and Links" by Dale Rolfsen (Publish
or Perish Press, 1976) (blanked out word will be obvious on solving the puzzle...). Bottom was done 'by hand',
although there was scope for some copy and paste, with reflections.
(February 2002) A crossover from ambigrams, this one is best viewed when printed out on an overhead projector
transparency, cut into strips, then made into actual Mobius strips, as in...
...this picture. There are 3 copies of the phrase 'Mobius Strip', but it is only printed 1.5 times.
(May 2002) Paul-Olivier tracked down a program (thanks to the author,
Toby Driscoll) which calculates the appropriate conformal maps (somewhat temperamentally!) and draws the lines in, then we just had
to colour in the squares. There is talk of doing a whole font in this style... Even larger version here.
(October 2002) This one is both an ambigram and a autologlyph. It's unfortunate that I
had to cheat with the first 'y', although at least it is recognisable as a reflected 'y' as opposed to some other letter. Maybe given the context I get a bit more leeway?
(December 2002) A bit of a departure here - I guess I could have done this as a word, the same way the other autologlyphs are, though that's not how the idea came up.
Maybe I'll do a word version as well.
(December 2002) A few new ideas here and the first ones from algebra. Representing a function as two parts, the domain and range, seems to be an idea worth
exploring. The third function down seems far too good to be true, but somehow it works. The lower left autologlyph needs to be printed
out and a hole punched in the marked spot to work properly. Here and here are 'answers' to two of them.
(December 2002) Fifth line down is a registered trademark, owned by Solomon Golomb. Based on an idea by Rick Rubenstein.
(December 2002) This was implemented by a program written by Neilfred Picciotto.
(January 2003) The images were generated by this freeware fractal generator. None of them are rotated at all, they're just copy and paste from the set.
The process of searching for letters is interesting: presumably because its a fractal, a good
technique seems to be to find something that looks near right, then zoom in around
it. Chances are something nearer to what you want is in there when you
zoom in, better orientation or shape, so I 'evolved' towards good
letters. Here is a bigger version on two lines for printing.
(January 2003) An ambigram-like idea. Ambigrams of two different words but without a rotation or other
transformation are very hard to do well - having a rotation makes it easier for the brain to ignore elements
of the image designed for one word when seeing the other. Here the 'transformation' is from words in English to
numbers and equations.
(January 2003) Based on a design by Will Segerman.
(February 2003) This was by far the most complex and ambitious project yet. The basic idea is the same as two of the autologlyphs in 'autologlyphs4.jpg', but we went much further with this one. The basic pattern is a tiling of the Poincare disk with five pentagons around each vertex. The vertex is now inside the tip of the 'E'.
Paul-Olivier worked out how to use this Mathematica notebook by Matthias Weber,
which deals with the calculations needed to do rotations in the Poincare disk and create the images. We put in many of the points of the central copy 'by hand', typing in coordinates. Many of the others needed to be
on geodesics defined by other points, and the notebook has tools to do that sort of thing. Having worked out how to get it to rotate words of a colour to other words of that colour, we had Mathematica output as pdf files the outlines of those words in each colour (so for example one file for red 'disk's).
This makes colouring them correctly in in Photoshop possible. From start to finish it took between 50 and 100 hours of work over a period of a month, with maybe 15 hours of computer calculation.
The final output is a huge gif file, 9600x9600 pixels and just over 4MB, so looks good printed at poster size 32"x32" at 300dpi. Available on request. For a slightly less ludicrously large image, look at this.
(January 2004) A similar idea to the Poincare disk, but was much less work, only needing a pentagon, printer, pencil, scanner and Photoshop. In the Poincare disk as in this the word(s) fit inside a pentagon, and in PD the neighbouring pentagons are related by a 180 degree rotation. Here only one of the 5 edges is like this, and the others pair up differently, which leads to some interesting symmetries over the whole dodecahedron. When folded up, it looks like...
...this. A high resolution version good for printing on letter size paper at 300dpi is here.
(May 2004) This was sent to me by Saty Raghavachary. (If more people send me autologlyphs I'll probably set up a separate section for others' work :)
(August 2004) Another reasonably large project, I first had the idea about a year ago. Paul-Olivier adapted Duane A. Bailey's Penrose tiling program, and used data from paths drawn in Photoshop, to make a postscript file to draw it. Then back into Photoshop for colouring and effects. The pattern extends...
...like this. In the other tesselation autologlyphs each edges matches to only one other edge, so the line between them has to serve two purposes. Here, because of the choices in how the two tiles can fit together, the two kinds of edge of the tiles have to serve four purposes.
Alternatives
! 1, a
bi Jr
odd tri
even four, quad
prime penta, fifth
phi(7) notodd, sextet, half12
seventh noteven, dwarves, wonders
twocubed twofours, powerof2
composite
triangular squarefree
twoplusnine fiveplussix, primenumber
zodiacnumber
superstitious
twoweeksofdays einhexadecimal, Februaryhalved
fifthtriangular
fourthpoweroftwo gjashtėmbėdhjetė (in Albanian)
foursquaredandone
fiftyeightmodforty
primeafterseventeen
positionofthelettert writeatwonexttoazero
sevenandsevenandseven
secondmultipleofeleven
smallestprimeovertwenty
twicetwelveorthriceeight
raisefivetothesecondpower areaofasquarewithsidefive
halfofastandarddeckofcards countslettersinthealphabet
multiplythreebythreebythree
theonlytwodigitperfectnumber theeighthprimenumberplusnine
daysinfebruaryifitisaleapyear
multiplyingthefirstthreeprimes
baskinrobbinshasthismanyflavors
takethesquarerootoftentwentyfour
trentetroiswhentranslatedtofrench writedowntwothreesnexttoeachother
twothirdsofonemorethanhalfahundred
multiplytogetherromannumeralVandVII
firstperfectnumbermultipliedbyitself
thinkofanumberbetweenoneandonehundred
inbinaryitbecomesonezerozerooneonezero
writethefirstoddprimenexttoitsownsquare
ninthwhencountingbyfivesstartingwithzero
itspositioninalistofprimesisseenasbadluck
hhgttganswertolifetheuniverseandeverything dadamsanswertolifetheuniverseandeverything
fourthprimewithleastsignificantdigitofthree
countbytensuntilyoureachfourandaddtwosquared
threetimesthesquarerootoftwohundredtwentyfive
numberofgoldmedalswonbytheUSin2004plustwosixes
wordsinthepreambleoftheUSAconstitutionminusfive
numberofhoursonananalogclockfacemultipliedbyfour
countyourtoesmultiplytheresultbyfiveandtakeoffone
thenumberofmembersofthejedicounciltimesfivelessten
.
.
.
anunreachablepointattheendofaneverendinglinethatrepresentsanunreachable.... (inspired by an ambigram by John Langdon)
The 'test question' for membership in this list is something like:
"Is the number of letters in 'foo' foo?". Something like... Suggestions to fill in the gaps or alternatives to the usual address.
Induction?
If n>8, and we have a word for n-8, [w] say, then "[w]andeight" is a word for n...thus by induction we now have words for
all natural numbers? Well maybe, here's a possible counterexample suggested by Paul-Olivier:
Contributors (that I can remember, tell me if I've forgotten anybody):
[lots of people from the Stanford Math department], Scott Sheffield,
Paul-Olivier Dehaye, Daniel Ford, Henry Segerman, Dave Futer, Elizabeth Meckes, Mark Meckes, Rick Rubenstein, NeilFred Picciotto (lots), Jeff Rissman, Ryan Stewart
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