Namesake :-)

M y friend & blogreader Regina tells me I have a namesake on Facebook :-)

So let me take this opportunity of greeting him & all of his friends & followers.

From the photo alone, its seems we have in common blue jeans and baldness [I have no pool though] ;-) But this screenshot and photo are all I have to go on, since I don't use Facebook myself, because I disapprove of what Facebook calls a 'Privacy' policy :-( Nor for that matter do I use Twitter or Myspace, for the same reason.

Here's wishing you well, unknown namesake :-)

But remember, today is Quit Facebook Day !

Stu Savory on May 31, 2010 permalink Comments Email

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Braveheart Anachronisms

I saw the movie Braveheart in the cinema when it came out. It is alledgedly a 'true' history of William Wallace, Scottish patriot, the director and the lead star claim. So when it was show on TV last week I decided to see just how many (ten?) anachronisms I could spot. I ignored all the continuity errors, there lots of those :-(

• Some people have tooth fillings: in 1300 AD?
• Blue jeans and even a back-protector are seen to be visible under the Magistrate's chain mail as he rolled down the hill.
• Some fighters in the battle scenes are wearing running shoes, and come to that even wearing modern wristwatches.
• The horse that William Wallace rides to the garrison wears horseshoes.
• Wallace's black underpants (under a kilt!!!) can be seen during the scene before Murron gets killed.
• Come to that, why are they wearing kilts? The kilt wasn't worn in Scotland in Wallace's time.
• When Wallace's wife is killed, a white Ford Transit van is seen in the distance.
• Apropos vans, after the Wallace/Isabelle love scene, a van can be seen far away at the bottom of the hill.
• At the real Barns of Ayr massacre, Wallace was 29, not a youngster as shown in the film. I guess we have Blind Harry (170 years later) to blame for that ;-)
• Wallace actually died in 1305. Isabelle did not arrive in England until 1307, aged just 12. So they are hardly likely to have been lovers. Lax 'history' there ;-)

Stu Savory on May 28, 2010 permalink Comments Email

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How to explain this 'lapse of good judgement'? It's claimed she DID have an exorcism a while back. But since she couldn't afford to pay for it, she got re-possessed ;-)

Stu Savory on May 26, 2010 permalink Comments Email

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Fractal Dimension measurement

L eon has asked how to measure Fractal Dimension numbers. The simplest way is to use the box-counting method. Look at the line segment on the graph paper on the left. Seen as one square, the segment would be in that one square. Seen as 4*4 squares, the line segment would be in six of them. Seen as 16*16 squares, the line segment would be in 17 of them. Go count 'em :-)

To calculate this dimension for last friday's map of Great Britain, imagine that coastal HiRes (4096*4096 pixel) image lying on a 4096*4096 grid, and count how many boxes contain a piece of coastline. The box-counting dimension is calculated by seeing how this number changes as we make the grid sparser (2048,1024,512,256,128,64,32,16,8,4,2 and 1 square). Powers of 2 deliberately chosen :-)

Writing the box size as ε pixels and (conveniently) using logarithms to base 2, we can calculate the Box-counting dimension as :- .

The easiest way for you to do this is to draw a straight line on log-log graph paper through the 13 points (there were 2¹² pixels) and then measure its slope. You should get fractal dimensions ranging from 1.02 for the coastline of South Africa to 1.25 for the West coast of Britain, even more for Chile and Norway (thankyou Slartibartfast ;-)

FWIW, this box-counting dimension is also known as the Minkowski-Bouligand dimension after the folks who came up with the idea. Theoretically, it is generally better to use balls than boxes, since the advantage of using balls rather than boxes is that this definition generalizes to ANY metric space. However, the advantage of using boxes is that in many cases N(ε) may be easily calculated explicitly as I just showed you. Caveat Emptor : Practical dimension estimates are VERY sensitive to experimental noise, and particularly sensitive to limitations on the amount of data; my 13 point example is TOO tiny really. I just chose it for blog-consistency.

There are other measures too, most notably the Hausdorff dimension, but the box-counting one is easiest for the layman to understand :-)

And before you ask WHY this is of interest to me : as a cryptographer, I can use the fractal dimension of an encoded (one-time-pad?) message to test whether it is truly a random sequence or has merely been constructed using a pseudo-random number (PRN) generator. If the latter, and the PRN has a fairly short cycle length, I can make a statistical attack on the message, maybe breaking the code. Maths IS useful ;-)

Stu Savory on May 24, 2010 permalink Comments Email

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What bugs me?

I am no entomologist, so if any of my readers can identify this little (finger-nail sized) fella with his impressive african-death-mask pattern on his back, I'd be grateful. I don't even know if it is indigenous here.

What kind of bug is it?

Comments (5) :
Peter Harris (UK) quips : "ho ho ho ,should not let it near your computer it might be a trojan from one of those african villages that support modern art!"
Friend Gerald (D) answers : "I can only provide the German name: Feuerwanze. "
Decrepit Old Fool (USA) also answers : "See Wikipedia on Firebugs". Thanks, George.
Old friend Jenny (Ibiza) claims enviously : "I don't remember the name, but I do remember they can screw for up to a week at a time ;-)" Sounds familiar ;-)
Brian (UK) quips : "It's a security hole in FireFox, hence called a FireBug ;-)" Groan!

Stu Savory on May 22, 2010 permalink Comments Email

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Measuring Coastlines, all of which are fractal.

C ommenting on tuesday's blog entry, Leon asked "Tell us about measuring coastlines & what about tides?"

Let us start at Land's End (A) and assume we have a measuring rod so long that the other end does not touch the coast again until point B (opposite Ramsey Island, Pembrokeshire, in Wales). Then our measuring rod is not short enough to reach into the Bristol Channel (up to point H), thus ignoring it entirely. Starting again at point B, our measuring rod reaches the coast again at point C (near Llandudno), ignoring the whole bay west of Aberystwyth. Between points C and D (near Stranraer, Scotland) all of the coast east of the Isle of Man gets ignored. Later the estuaries E,F and G get ignored too. Obviously, the shorter our measuring rod, the longer the result, and the nearer it gets to the 'true' value (whatever that turns out to be [ask the CIA ;-)] ).

Rod size matters ;-)

With a measuring rod 200 km long our result would be about 2400kms. For 100 it would be 2800, for 50 about 3400 kms etc etc. The CIA factbook gives the length of the UK coastline as 12,429 km without saying whether they included all the islands or indeed how long their little rod is :-( It turns out that the coastlines are statistically self-similar and have fractal dimensions ranging from 1.02 for the coastline of South Africa to 1.25 for the West coast of Britain. Fjord-heavy Chile and Norway are other extreme cases which we owe to Slartibartfast, a planetary coastline designer who was responsible for the fjords of Norway (see HHGTTG q.v.) ;-)

In all of this I merely consider Mean Sea Level and ignore tides, let alone individual grains of sand and atomic level phenomena.

Of course, with a real (mathematical) fractal the perimeter could be infinitely long although the island has only a finite area. Let's consider Koch's snowflake as a simple example. Take an equilateral triangle. On the centre of each side construct an outward-pointing equilateral triangle. Note that - although these new triangles are smaller - the perimeter (of the star-of-david which we now have) is now 4/3 of the length of the initial perimeter. Continue doing this forever, the perimeter increases by factor 4/3 at each iteration, reaching infinity. The area remains finite though : 8/5 of the area of the original triangle. The Koch curve is continuous everywhere but differentiable nowhere!

Comments (4) :
Leon (NL) is still curious "You wrote : '..... have fractal dimensions ranging from 1.02 for the coastline of South Africa to 1.25 for the West coast of Britain' : can you tell us how we (not you) can measure fractal dimension numbers?" Coming up, Leon :-)
Jenny (Ibiza) asks coquettishly : "How far can you put your (short ;-) rod into the estuaries for them to still count as coastlines? I understand it depends on the size of your rod, but where is the river/coastline border?" Hi Jenny babe, long time no sea see :-( Dunno how they defined that. With a short enough rod, you could go all the way up to the source of the rivers, I presume. Or maybe it's the mean salt-/fresh-water limit? But what about the Severn Bore? I just don't know, sorry :-(
Kees Kennis adds to my knowledge of diverse trivia, with this :- "You can add the Gambia River in The Gambia. Gambia's highest point is 42m above MSL and the river is salty for 80 km inland. If you measure The Gambia's ratio of Square Km to Km of Shoreline I think it would be amongst the smallest."
Stu adds an off-topic note to BP : Fractals can be used to model oil spills leaks too :-(
Babs (US) asks "What is the fractal dimension of that snowflake?" It's 1.26, Babs.

Stu Savory on May 21, 2010 permalink Comments Email

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Superficial Story

W ay back in 1962, in my first week at university, reading Physics, we had our first lab session, which was quite insightful for me. Here's the tale...

We were given a regular incandescent lightbulb and asked to measure its surface area. We were also given a pot of paint, a ball of thread, calipers, a balance and a micrometer as tools.

My first idea was to measure the exact weight and diameter of the bulb, then dip it in the paint (suspending it from the thread) and let it dry. Afterwards reweigh and remeasure the painted bulb. From the diameter difference I would know the thickness of the paint. From the weight difference the weight of the dried paint (whose density I could get from the weight and volume of a separate sample) I could calculate the surface area. I had to reject that idea; it would have taken longer for the paint to dry than the two hour duration of the lab session available to us :-(

So I ended up sticking the thread closely along the bulb surface top to bottom, using dabs of paint to hold it in the screw thread at the bottom. I now knew the length of thread needed for the lengthwise semiperimeter of the bulb. I then marked the thread in 1 cm intervals with dabs of paint and measured the diameter of the symmetrical bulb at each 1 cm mark (calipers, micrometer). Thus I got an average diameter for the bulb. Given this and the thread length I could do the math to calculate the surface area.

But although I got a B for effort, I only got a C for my solution :-(

My bright friend John did nothing, turning in a single sheet of paper on which was written "Surface Area is an ill-defined term. Any result measured will depend upon the roughness of the surface being measured and whether you measure over or into the roughness of the surface. Traditional Greek surface area is defined as a minimum measure for an ideal two dimensional surface. Real surfaces are three dimensional."

John got an A :-)

The lab lecturer said "At school you learned by doing what the teacher told you. At university - and later in real life - you must always first ask if the question is sensible and the terms used are well defined and commonly understood by the questioner and the answerer!" That meta-lesson served me well during my days as a consultant :-)

But if you look closely, you see how close we were (in 1962) to the concept of fractals, but as freshmen we had neither the time nor (yet) the mathematical ability to pursue that idea to its logical conclusion (fractal dimensions). In fact the seminal paper on fractals did not appear until the year following year our graduation (1967). How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. Here is the Wikipedia article. Mandelbrot built upon significant earlier work by Lewis Fry Richardson, to whom more credit is due IMHO. The word fractals was not coined until 1975.

So near and yet so far!

In retrospect, this story shows how important it is to think things through thoroughly!

Comments (1) :
Leon (NL) asks "So tell us more about measuring coastlines. What about tides?"

Stu Savory on May 18, 2010 permalink Comments Email

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Evolution of the English Bulldog ;-)

T his blogpost is for Kees Kennis, who believes a lot of weird things and who has been teasing me gently recently about my love of 'German' bulldogs, as he calls them (merely because I live in Germany).

If we were to follow Kees's theory we would have to assume that the English Bulldog evolved from lions on the plains of Africa (where Kees lives). So I've made an animated GIF which illustrates the evolution for Kees ;-)

OK, OK; well at least the colour didn't mutate; you can't win 'em all ;-)

Copyright ^© Stu Savory on May 16, 2010 permalink Comments Email

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Just do the Math :-)

Pull out your pocket calculator, or pencil and paper, or tax your mental arithmetic capabilities, but just complete this little pyramid here. Kinda neat result :-)

Copyright ^© Stu Savory on May 14, 2010 permalink Comments Email

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One down, 665 to go :-)

Some good news at the weekend : the Pope has accepted the resignation of Bishop Mixa of Augsburg who is currently accused of a) beating small children, b) sexually abusing children, and c) fraudulently misappropriating church funds. Police are investigating :-)

I'm just curious : I wonder which of these three allegations got him fired?

Incidentally, have you noticed that neither the Pope, the bishops, nor any other Catholics ever pray for Satan, although surely He is the one who needs it most? ;-)

Copyright ^© Stu Savory on May 12, 2010 permalink Comments Email

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Prude's news ruse :-(

Wikipedia has always mostly been dedicated to providing educational materials online. But now Fux News Fox News has bullied them into censoring their website based on their prudish merkin view of 'things' :-(

Of coarse (sic!), what you deem to be pornographic depends on the particular culture you live in; e.g. fundamentalist Islamists disapprove of anything less than a full burka. And Wikipedia is available world wide, so why should Fux Nudes Fox News get to impose their particular uptight views on the rest of the world, yet still object to the Islamists doing the same?

I wonder how Fux Nudes Fox News sees the little icon at the top of my blogpost today? Do they see the Japanese temple in front of the setting sun? Or do they just see a symbollock (sic!) doggy-position penetrating icon ;-) I bet it's the latter ;-)

Copyright ^© Stu Savory on May 11, 2010 permalink Comments Email

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Jock Rock :-)

Our local city - Paderborn - held a spring festival over the weekend and kicked off on friday evening with a free open air rock concert. The main band - playing 9-11pm - were the Scottish bagpipe rockers The Red Hot Chili Pipers.

Three drummers (2 mobile, 1 sessile), a keyboarder, three pipers and the hardrockin' guy with his trademark red mohawk haircut on guitar. We managed to get up to the front - deafening - where the martial beat of the drums and the skirl of the pipes probably exceeded 120 dB. My ears rang all the way home, so it was really great :-)

Drum solos, guitar solos and of course bagpipe solos interspersed the main group numbers to give the rest of the band a breather. Occasionally, the pipes' drones got out of tune ("How can you possibly tell?" as the English often ask ;-)), but a quick re-tuning twist of the drones soon put a stop to that.

Popular rock numbers played in their own Bagrock style included the well known Deep Purple's Smoke on the Water, Queen's We will Rock You and Jimi Hendrix's Voodoo Child.

Traditional numbers included Hills of Argyll and my favourite Flower of Scotland :-)

Copyright ^© Stu Savory on May 10, 2010 permalink Comments Email

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No fair election :-(

A while back, Britain handed Hong Kong back to China. And since the UK voted yesterday, a Brit named Brian called his old Hong Kong friend and asked him "Do you have elections in China?". Said Hong Kong friend grinning into the video conference call replied "Yes, Blian, evely molning!" :-) . . . . I'll get my coat ;-)

Having cracked that one, let me tell you about Kenneth Arrow's 1963 proof that there is no such thing as a fair vote! Kenneth Arrow (an American economist) listed four basic properties of an idealised fair voting system :-

• Voters should be able to express a complete set of their preferences; this means ranking all the candidates, not just selecting one of them.
• No single voter should be allowed to dictate the outcome of the election (ie no kingmaker minority).
• If every voter prefers one candidate to another, the final ranking should reflect that fact.
• If a voter prefers one candidate to a second, introducing a third candidate should not reverse that preference.

However, Prof. Arrow (and others) then also proved that no conceivable voting system could possibly satisfy all four conditions!!!

For those interested, here's the proof, (by Geanakoplos, Yale, revised 2001, PDF file).

In the UK, people were turned away from polling booths which couldn't cope with so many voters turning up in the last 2 hours before the rigid 10pm closure deadline. Elsewhere, students were put in slower-moving lines, preventing some from voting before the deadline. Vote-rigging by Bangladeshi immigrants has been reported too. It seems that the new parliament will be (well) hung [Balls re-elected ;-)]. This means that Gordon Brown, despite losing heavily at the polls and not being the largest party, still gets first dibs at forming a new government :-( Call that a fair election?

FWIW, not only did the UK vote yesterday, our state (NRW,Germany) votes sunday.

Copyright ^© Stu Savory on May 7, 2010 permalink Comments Email

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NSU R54 info sought : your help needed please

RFI If you have any information, technical, anecdotal, photographic, acoustic or on film, about the extinct 1950's NSU four cylinder inline racing motorcycle type R54, please send it to me and I will forward it to the interested parties. And thereby hangs a tale . . .

20 years ago I researched and wrote a novel about motorcycle racing in the fifties. It is the one shown on the right above, entitled "Howl of the Mountain King". It is now out of print. Needless to say, I never owned any of the works machines described and all my information was second hand [anyway, I would have been about 12 at the time the novel describes]. Nevertheless, I did my best to get technical and historical details correct, except that the hero's square four was based on the Suzi RG500 of the early 1980's.

Now, 20 years later, and out of the blue, I got an eMail from Ray Battersby, author of the authoritative 1980's history of Team Suzuki, shown on the left, above. Ray is looking for any and all information about the NSU R54, of which very little is known. Ray had used the power of the Internet and its search engines, and one of the few hits about the R54 was from a chapter of my novel, whence his questions about what else I know about the machine. Alas nothing else, and I no longer have my novel-writing notes nor any photos. Two decades have erased my memory too:-(

Aside: My novel was about East German two-stroke expertise. When Ernst Degner^* defected, he did so to Suzuki, and was debriefed by them about Walter Kaaden's MZ tuning. Subsequent Suzukis used much of the MZ technology, I think it is fair to say ;-) So this would be a point where Ray and I overlap, without either of us knowing it at the time we wrote our respective books [Ray's book is from the 80's, I guess] :-)

Anyway, it turned out that Ray was asking the questions for Kevin Cameron, renowned author of several motorcycle books (two are shown below) and a regular contributor to Cycle World. He is probably the world's most widely read motorcycle author. If you , dear blog-readers, can help him in his researches, it would be great :-)

Turns out Kevin had a copy of my novel on his desk and was reading about the R54 in it. It took a while at my end to realise I was conversing with THE Kevin Cameron - shame on me - as opposed to Kevin Cameron the baseball player, or K.C. the Oregon State Representative, or K.C. the director of Reddy Ice, or K.C. the . . . [thanks to Charles Pergiel for the list ;-) ]. But Kevin is a great conversationalist, and we've been swapping nostalgic motorcycle stories etc. etc. Here is a sample :-

Stu : ". . . when at university (City University, London, UK, B.Sc (Hons) in Physics, class of 66) I really wanted to excel at motorcycle racing instead. I had a 250 Ducati Mach 1 (on triangular tyres!) - like Chas Mortimer - which I used for club racing (Brands Hatch etc). Turned out I was only competitive at club level, thrashing the Greeves Starmakers etc. :-( Couldn't afford anything faster :-( On Brands practice days Bill Ivy (works Yamaha) would run rings around me, coming past, frame snaking, steering head twisting, boot sparking. When I borrowed Vic's 7R, Derek Minter would come blasting past me on the inside, braking 20 yards later than I :-(
I then realised I was too cowardly to become a real roadracer ;-)

So I built a sprinter, C15 on methanol in a cut down Bantam frame. Uncompetitive heap'o'crap, but cheap! Meanwhile Alf Hagon was running under 10 secs even back then (62-68)... Those were the Super Nero days... So I built a blown Norvin . . ."

Kevin : " . . . I could tell you had a solid background in all this ;-)

The high point of my brief attempts at road racing was the weekend I had just picked up a set of Triangulars at the airport, courtesy of a friend who'd air-shipped them to me. With those on my Yamaha TD1-B I found myself in the usual approach conflict (I want the food pellet, but I don't want the shock that usually accompanies reaching for it) in Gunnery Corner, Harewood Acres, Ontario, behind one of those loud Ducati 250s. Suddenly I thought, "I am on a set of the best tires in the world. I must get going". And so I turned the grip, pulling up on the Duc. Its rider, hearing the awful sound bearing down, opened further himself, and leaned over farther (we were both lowly novices, mind you). Soon sparks poured from his peg, his bike rotated, and he slid out of my frame of vision. I felt as though I had destroyed the enemy - a tremendous, rather embarrassing elation. Spittle may have dribbled down one cheek from my open mouth. Modern helmets save one from being seen thus.

Later I had the pleasure of working with the late Cliff Carr, who rode my Kawa triples 1971-73 in the US. I never had any notion of writing as a means of daily bread, but that's how matters turned out . . ."

I've pointed Kevin to Paul d´Orlean and Wolfgang Schneider and Winni Scheibe (who lives only 30 miles from me), to Paul Gockel (Bonham's man in Germany), Friedel Münch and Inge Rogge (for her photo archives). But if any of you esteemed blogreaders have further R54-relevant contacts, please mail them to me; Kevin will be grateful, I'm sure, and you may even get a mention in his column (and this blog) ;-)

Comments(1) :
Here are two Karl-Heinz Bendix R54 photos found online, here and here.

^* Ernst Degner died of a heart attack afaik, so let's stop all the other silly rumours please!

Copyright ^© Stu Savory on May 5, 2010 permalink Comments Email

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Nukulah Deterrent :-(

J ust a preliminary remark as we head into Cinquo de Mayo :-

The USA revealed yesterday that it has only 5118 nuclear warheads now left [that is my emphasis]. I wonder how many of them helped prevent the Nashville Flooding or the Gulf-of-Mexico Oil Leak or the weekly collapse of a ½ dozen US banks?

Comments (2) :
Charles Pergiel (USA) quipped " Just imagine how much worse each of those disasters would have been if we had only 4,000 nuclear warheads. It would have been catastrophic! " My point exactly, Charles. How much Health Care, Green Energy, National Debt Reduction could the US afford if it halved its military expenditure? :-)
Jedi Night(sic!), USA, wrote : "May the Fourth, be with you ;-)"

Copyright ^© Stu Savory on May 4, 2010 permalink Comments Email

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Read the Classics!

P eople sometimes mock me when I claim that a classical education^** can't hurt you. But here's a recent story from the UK press which supports my belief ;-)

Those of you well versed in Greek mythology may remember the story of Ariadne's thread; for the rest of you I'll string together[sic!] a brief synopsis :-

Ariadne was daughter of King Minos of Crete. She helped Theseus defeat the Minotaur. The Minotaur was a creature with the head of a bull on the body of a man who lived at the centre of the Cretan Labyrinth, a maze built for King Minos of Crete by the architect Daedalus and his son Icarus (he who flew too close to the sun, remember?). Theseus had the problem of tracking into the maze, killing the Minotaur and getting back out of the maze without getting lost inside forever like the Minotaur himself.

To this end Ariadne (who was smitten with Theseus) lent him a ball of red thread which she was spinning. Theseus used the first recorded example of a back-tracking algorithm, viz track through the labyrinth taking left turns wherever you can, paying out Ariadne's thread as you go. If, at a junction, you encounter the thread again, take the right fork instead. This will take you all the way through the maze if necessary^* and you can follow the thread back out.

This was such a popular story that a song was written about it : A Maze in Greece ;-) Sorry about the pun, I couldn't resist it . I'll get my coat . . . ;-)

Anyway, to get back to the recent story from the UK press : turns out that a spleeny Brit - they just love messing about in boats - wanted to move his motorboat from a mooring well up the Thames estuary through the English Channel to a mooring at Southampton on the south coast. Being a spleeny Brit, he set off without any nautical maps or navigation equipment of any sort, just a road-atlas large scale map of the coast!! Now just a few miles eastwards in the Thames estuary he encountered the Isle of Sheppey, presumably in rain and/or poor visibility, and followed its coastline!. Sheppey being an island, he proceeded to circumnavigate it several times (having no knowledge of Ariadne's thread) until he ran out of fuel and had to be rescued by the Coast Guard :-) Sky news has the full story here. Read it and die laughing ;-)

On a not unrelated thread(sic!), if you are interested in messing about in boats, you could do worse than read the new book by my friend Friedhelm Würfel.

Mathematician Friedhelm and I worked together some 20-30 years ago; at weekends he would sometimes take me sailing in an unsuccessful attempt to get me enthusiastic about 'messing about in boats'. In this family, my wife is the one with the skipper's licence, both literally and figuratively speaking ;-)

Anyway, Friedhelm has written (and photographed) an autobiographic book relating how he built his own Alaskan Skiff and then travelled north up the coast of Sweden in it. Lots of beautiful colour photos, which I would have preferred to see even larger, but he had to leave some room for the text ;-) The ISBN is 978-3-86805-623-5.
More details about Friedhelm and this book, available via www.friedhelmwuerfel.de :-)
Oh yes, I forgot to say, its 176 pages, A5 landscape format, are written in German.

Comments (2) :
Petra (D) came up with a limerick :-

"There once was a mathematician
who built his own boat to go fishin'.
He took pictures instead
wrote the book you just read
and thought this a real good decision."

Fellow geocacher Gerald (D) sent me "some really nice sea stories, too: like these, and this one (Eric Abbott), or this one(Eric Abbott again), and even this one (even more about Eric Abbott). ;-) " Captain Calamity indeed, Gerald! Thanks for the links.

^* Actually this algorithm does not work if the maze contains islands, e.g. has the connectedness of a small doughnut, coincidentally known to (Greek) mathematicians as a mini-Torus (pun intended).

^**Si Hoc Legere Scis Nimium Eruditionis Habes ;-)

Copyright ^© Stu Savory on May 3, 2010 permalink Comments Email

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