The Haunted BBC Micro?

I used to have an Acorn Electron. The thing about Electrons is that they think they’re BBC Model B microcomputers. Their system software is pretty close to or actually identical. However, when you come to actually use them, it becomes clear that they aren’t. They lack MODE 7, the Teletext mode, only have one sound channel and only have an edge connector as an interface. The CPU running both models of computer lacks specific I/O ports, unlike the Z80, and therefore peripherals have to be mapped directly onto the memory. Due to the hardware shortcomings of the Electron compared to the BBC B, there are unused spaces in the memory where the interface chips would’ve been.

One day I was looking through the Electron’s ROM (system software) and wrote a program to print out the printable bits of these regions. If you just look at memory contents and output them as characters, you end up changing the graphics mode, position of the cursor and so forth, and the colours on the screen, and while that’s entertaining for a bit it isn’t conducive to actually finding out what’s in the computer. This is because the ASCII control characters don’t actually print, and the BBC/Electron version of the character set is substantially used to communicate with the display hardware in quite sophisticated ways, probably because the BBC hardware is supposed to be adaptable as a terminal for the second processor. This second processor was ultimately to be the famous ARM whose descendants run today’s mobile phones. Hence the BBC is very much about telecommunications in that sense as well as many others. Anyway, if you blank out the most significant bit of the bytes in the memory and also only print out values above 31 (1F in hexadecimal), every character written to the screen will be printable. If you then look at the area of memory which is used on the BBC for peripherals, you find a list of acknowledgements for the people who designed and built the Acorn Electron. For some reason it isn’t stable and the longer it’s been since the computer’s been turned on, the less legible it is, so it’s a race to get to see it, but it’s there. I don’t know why it degrades. Doing a reset doesn’t restore the data either: you have to turn it on again to do that.

I seem always to back losers. For instance, I was a Prefab Sprout fan. If I like something it’s the kiss of death for it. Therefore, unsurprisingly, as well as the unsuccessful Electron I also had an even more unsuccessful Jupiter Ace. I used to do something similar with the Ace’s memory, dumping it to the screen. This is simpler with an Ace because it has fewer control characters. The 3K of static RAM in the Ace, as opposed to the dynamic RAM in the RAMpack, has a load of apparently random values when you turn it on, although like any other computer it also has system variables, and like many others it has working areas, video RAM, character shapes and the famous PAD FORTH uses for text manipulation, and of course the parameter stack as it’s a FORTH computer. The dynamic RAM of the RAMpack has blocks of zeros and hex FFs (255) in eights, I think, all the way through the unused map, which I assume to be an artefact of the hardware, although presumably the CPU does that thing of writing bytes every 256 locations or so to work out how much memory the computer has. Every time an unexpanded Ace is turned on, it has the same junk data in its RAM.

This phenomenon of nonsense in RAM and defining a word which displays it on the screen gave me the idea I hold to this day of the nature of dreams. It would be possible to get an Ace to turn those data into words. I’ve got it to produce random words in Finnish, for example, mainly because Finnish is an easier language to get a computer to produce than almost any other. English is a lot harder. I could’ve linked the two things together and got the Ace to turn all its random data into Finnish. I didn’t do this because I decided to go cold turkey on computers in about 1985 because I don’t trust my own interests and they seem a bit obsessive and unhealthy, but if I had, I wonder if it would’ve produced different Finnish for every Ace in existence, or if the random data were the same for all Aces. It didn’t happen on the ZX81 by the way. That just has zeros all the way through its unused memory. Anyway, this is my hypothesis about dreaming. When you wake from a dream, your memory contains random data like an Ace’s memory, and your consciousness is like the Finnish converter. It attempts to make sense of these data and you get the impression that you’ve just had an experience, although you usually know you haven’t. This is one of the reasons I always refer to events in a dream in the present tense, because the events in them did not happen in the past. However, this shouldn’t be taken to mean that they are invalid. Dreams are like tea leaves. They can be interpreted as a way of approaching reality with the added benefit that they’re already partly in this state when we receive them.

In I think February 1984 CE, economics teacher Ken Webster took a BBC Model B Micro home from his school to his seventeenth century cottage in the Cheshire-Flintshire border village of Dodleston. I’m going to be fairly brief about the details of this case, which is extensively written up elsewhere, including in his book ‘The Vertical Plane’, because I want to concentrate on something else. There were three people in the house: Ken, his girlfriend Debbie and a musician who lived upstairs whose name I can’t remember. That night, he left the computer on and the house was vacated when they went to the pub. On coming back, a poem had appeared on it. Over the next sixteen months a series of messages appeared to which he and some other people responded. Here are a few screenshots from a dramatic reconstruction:

I shall explain. There was an apparent dialogue between Ken and Debbie and a person appearing to live in the sixteenth or seventeenth century called Tomas Hardeman (living in the time before standardised spelling so his name is uncertain) who initially claimed to be a graduate of Jesus College Oxford and later Brasenose. There are both historical and grammatical inaccuracies in the messages purporting to be from the past. Tomas Hardeman is arrested for witchcraft and only released after the Ken threatened the sheriff who arrested him, who was apparently also communicating. The messages are then interrupted from a source known only as “2109”, possibly a year, which is more threatening and claims to be made of tachyons. Its spelling is also a little peculiar, with single consonants where we might put double ones and the “-tion” ending being spelt “-cion”. At the same time, there was poltergeist activity in the house, particularly the kitchen, where utensils tended to be piled up, and on one occasion Debbie came back to the house to find the cats nervous and all the furniture piled up in one corner of the living room. Brasenose College helped with the research and it emerged that there was indeed such a person who was expelled from the college for refusing to remove the Pope’s name from certain books in the library, which confirmed what had appeared in the messages. The Society for Psychical Research (SPR) then got involved, typed a number of questions into the computer without disclosing them to anyone, sealed it in the room for an hour, then deleted the messages, and got a reply which implied that “2109” was aware of their content. David of the SPR proceeded to ask the “entity”, if that’s what it was, the solution to Fermat’s Last Theorem, which was only found in 1994. It replied that the answer was only to be given if the questioner was prepared to lose soul, mind and body, so they didn’t proceed. “Harman” then said that he would write a book about the events to prove that they had happened and hide it somewhere, so that when it’s found it will be demonstrated that this was not a hoax. 2109 mainly seems concerned not to cause a temporal paradox. Oh, and the house was on a ley line, but then so was mine so that’s not unusual. Harman also mentioned that his house was made of red stone, and foundations of a building made of red stone were later found in the garden, so the house which stood there before was like that and Harman complained about the alterations made to the house in the intervening four centuries.

The mistakes made in the grammar and history were attributed by “Harman” to interference by “2109”. Both the SPR and more general sceptics agree that it was a hoax, but Ken and Debbie, particularly Debbie, insist that it wasn’t and it’s still unclear how it was done. Debbie has been very up in arms about it and expressed her annoyance at being accused of faking. She said she couldn’t understand why people thought so because she was not motivated to do such a thing. There are, however, textual similarities between Ken’s own writing and Harman’s. For instance, 26% of nouns are preceded by adjectives in both sets of text compared to an average of 32% taken from contemporary texts composed by other people, and in Ken’s case the sample is very large as it consists of his entire published book of 374 pages. Although this seems like more than a coincidence it doesn’t rule out the possibility that he was either doing it unconsciously or that the poltergeist was associated with him in some way, but I’m still basically convinced it was a hoax. Nevertheless there are some enormous difficulties in explaining how it was done.

I’ve seen some annoyingly naïve descriptions of how this was done, so I’ll go into the situation as it was then. Both the internet and email existed at the time. However, although it would be possible to connect a BBC micro to the internet (not the web of course) or to a Bulletin Board System, this computer was not connected in this way. BBC micros do have local area network connectors in the form of Econet, but again this one was not connected, at least while it was in the cottage. The SPR suggested that signals were being sent along the earth line of the plug and socket through the wiring of the house. Other than ROM, this BBC had no persistent memory. As it happens, this particular model was being used to run EDWORD, a sideways ROM for word processing, at the time. It was linked up to a green screen monochrome monitor, presumably without a Faraday Cage, and there was a 5¼” floppy disc drive with discs available.

The frustrating thing about the investigation is that as far as I know, nobody seems to have examined the hardware involved. The fact that the monitor was presumably unshielded means that it would’ve been possible to detect the signal and read what was on it from nearby using a scanner of some kind, so the content of the questions the SPR guy typed wouldn’t have been secure by the standards of the time. There was a dialogue, or at least it appeared to be interactive, and although the BBC micro could easily run a program like the Rogerian psychotherapist simulation ELIZA or the paranoid “patient” Parry, the sophistication of the responses means it has passed the Turing Test, which would be quite an achievement for a 2 MHz 6502-based micro with 32K RAM and the same ROM.

I regard all this as a puzzle to be solved by naturalistic means, because of the grammatical and historical errors. For instance, in the screengrab at the top, “BEHALTHE” is a spelling mistake which would never have been made by an English speaker of that era, and “WOT” is also incorrect because Midland English at the time strongly distinguished “WH” and “W” in speech, although Southern English didn’t. These would’ve been easy to fake and they seem to be poorly faked. There is, however, a claim that 2019 had a hand in the apparently older messages, which would explain the historical and linguistic inaccuracies. It’s also likely to be a valid excuse that telling the SPR the answer to Fermat’s Last Theorem would cause a temporal paradox, although it could presumably be stored in a sealed envelope and the people could be sworn to secrecy. But I think strong corroboration of backwards time travel would lead to a paradox anyway, meaning that there could only ever be vague references easily refutable or impossible to corroborate, so this is exactly what one would expect from a responsible message from the future.

The idea of the earth pin is interesting. Although it seems to have been suggested ignorantly by someone who didn’t know much about computers, it would in fact be possible with some hardware modification. The back of the BBC Micro looks like this:

Power is on the right, and likely to carry an earth line. Even if it doesn’t, one could be used. One of the other interfaces could be connected up to the earth, although I’m not sure which would be best. The cassette port is able to transmit data at 1200 baud along a single line, so wiring the in and out to the earth internally and having a way to switch remotely between the two is possible. Alternatively, a faster connection could be made between the Tube and the earth, and depending on how the Econet works that might be another option. The RS423 is, however, the obvious choice as it’s a communications interface. There would then need to be something connected to the wiring of the house, possibly something like a radio mike, which could then transmit and receive to another computer or terminal fairly nearby. But all of this would obviously involve modifying the hardware inside the case. The presence of a sideways ROM makes it feasible, although Edword would then have to take up less than 16K to allow for the software. Having said all that, I think the comment about signals entering and leaving via the earth is probably just a sign of being uneducated about computers.

The reason for this explanation is of course the need to look for a cause other than communication with someone living several centuries ago and an entity apparently 124 years in the future. The other options seem to be that there was communication with an entity in the future, communication with a timeless entity or communication with someone living in the past and someone else living in the future, or just talking to a ghost. “Harman” mentions a “boyste” of “leems”, I think in his fireplace or chimney, which could be the computer itself or something else. It’s also possible that voice dictation was supposed to have been used at his end because of how it’s described, factually or not. “Leem” means a glimmer of light and “boyste” appears to mean box, which could refer to a CRT monitor. It feels rather away with the fairies to say this, but it was possible to dictate to microcomputers at that time, although I suspect it didn’t work very well. When I say “at the time”, I mean the 1980s.

It really does seem like a hoax, and the biggest issue is really how it was done. Although I’ve mentioned one feasible way, there could be others, and it makes more sense to seek an explanation in hardware hacks than the supernatural or time travel. But that doesn’t mean that there is no paranormal or time travel, and the poltergeist isn’t explained by any of it.

The Apple Mannikin

Back in the 1970s, computer graphics were at a relatively primitive stage. A lot of them were just wireframe, and this very style became iconic of high technology and the futuristic. The 1979 Disney Film ‘The Black Hole’ was notable for having the longest ever CGI sequence in a feature film up until that time, at around a minute and a half. Here it is:

In the cinema, that looked pretty impressive to me at the time, as I’m sure it did others. However, CGI as we’re familiar with it today also existed, as with NASA’s sequence illustrating the Voyager missions, which was however updated with textures from the mission itself. Then there was Sunstone, also from 1979:

A few years later, there was ‘The Works’ in 1984:

However, by then they should’ve known better, because changes were taking place in mathematics which were reaching some kind of climax by that point, namely research into fractals.

I don’t really understand calculus, but I probably inaccurately think of it in two ways: trying to work out where a wiggly line will go next, and finding the slope of a curve at a particular point, with the emphasis on “point”. It’s where my understanding of maths runs out and therefore a bit of a locked gate for me because of what lies beyond in terms of its practical applications, which I can’t access. Nonetheless I am aware that in 1872, Karl Weierstrass announced his discovery of a function expressed by a wiggly line on a graph which was spiky everywhere, no matter how close you zoomed in on it. This is of course the Weierstrass Function, and looks like this:

The zoomed in bit is to show that it’s spiky on every level. Although it’s a line, there’s no curved or straight stretch anywhere along its length where it isn’t changing direction, no matter how small the difference between the values of x is. This is referred to as “nowhere differentiable”. The function can be expressed thus:

where α=the natural logarithm of a divided by the natural logarithm of b. There are plenty of discontinuous functions like this, but this has values at every point. Sometimes it seems like the nineteenth century consists largely of the eighteenth century status quo and simplicity being overturned at every point, just as the seventeenth century feels like a time of rising sophistication after the relative calm of the sixteenth, preceded by the complexity of the Middle Ages, and so on, which of course makes sense from a Marxist and Kuhnian perspective (note the singular).

This was the first of a series of curves, infinite really, which became known as fractals. The standard, and wrong, way of describing a fractal is that it’s self-similar. There are many self-similar fractals, such as the Koch Snowflake:

This starts out as a triangle, to whose sides spikes are added, making a partly concave dodecagon, to whose sides spikes are added, making a four dozen-sided shape and so forth ad infinitum. The above shape, partly blurred by the fact that it isn’t a vector image due to the difficulty of using vector graphics on WordPress, has seven iterations and therefore 12 228 sides, or it would have if it was actually drawn as opposed to being a raster image. And we’re back to computer graphics. However, most fractals are not self-similar in that way. The coastline of this island is fractal. The shorter the ruler used to measure it, the longer it gets, and you could be reduced to measuring between the grains of sand on a beach or the bumps on a cliff face, at which point the tides and whether something counts as wet come into consideration, but it isn’t self-similar. There aren’t lots of “little Britains” just off our coast which themselves have littler Britains off theirs and so on, appealing though the idea might be.

A fractal is actually a shape with a non-integral number of dimensions. Whereas a square has two dimensions and a cube three, and a line one, it’s useful to consider dimensionality as having values in between whole numbers. The Koch Snowflake, for example, has about 1.262 dimensions, and Great Britain 1.21. The reason the number of dimensions a fractal has is not integral is that the “size” of some shapes, such as the measure polytopes of the line segment, square, cube and tesseract, can be thought of as its measure to the power of the number of dimensions it has, and this is in those cases a whole number but in the cases of fractals. The Koch Snowflake is a wiggly line which meets itself, but it comes close to filling the area around the perimeter of a roughly hexagonal shape, so it’s neither one-dimensional – it isn’t a line – nor two-dimensional – it isn’t a hexagon or a star – but somewhere in between. However, although these ideal platonic shapes are self-similar, most fractals are not, but that doesn’t stop them from having a fractional or irrational number of dimensions.

The real world is not like the smoothness seen in computer graphics, particularly earlier ones. The three videos at the start of this post are all coolly mathematical and, while difficult to produce, involve simple shapes textures with simple textures. With the aid of fractals, it became easy to generate this kind of picture:

This image dates from around 1982. In ‘The Works’, there is some kind of bumpy terrain and I’m not sure how this was generated. As far as I know, this was first used in a feature film, ‘Star Trek II’, in 1982:

The structure of this clip is quite interesting because it goes from old-style wire frame models through textured rendering of three-dimensional objects and ends with the mapping of a fractally-generated surface. At the end of the Voyager missions to Saturn in late 1980, it was mentioned that the CGI people who had produced the videos of the mission and mapped the textures taken by the Voyagers’ cameras onto models of the planets and moons had left to work on ‘Star Trek II’. I presume this is what they went on to do. Incidentally, this disbanding of the team working on the Voyager projects, which was related to the six-year gap between the Saturn and Uranus encounters, shows the difficulty the kind of societies which send rockets into space have with achieving long-term projects. They couldn’t just keep these people on the payroll for six years while they did nothing, so we get this clip but at what cost? What else didn’t we get and who else was “let go”?

This is a “making of” video of the same:

A further tangential detail: the star field is as seen from ε Indi. Alnitak, Alnilam and Mintaka are seen as lined up near the beginning of the clip, indicating their relatively great distance, and as the commentary mentions, the Sun is visible as part of Ursa Major near the end. The constellation of Indus is opposite that of Ursa Major in the sky – it’s a Southern constellation – and ε Indi is almost twelve light years away. This particular sequence is a milestone in the development of CGI.

Raster scan CGI on flat displays is often quite rationally organised at a fairly low level, in that the screen is seen as a rectangular array of pixels like a graph, with the origin either at a corner or the centre. This means that the famous Mandelbrot Set image – the Apfelmännchen or “apple mannikin” as it’s known in German – is effectively a graph with the X axis running horizontally along the middle of the picture. It’s often difficult to remember that this X axis at the centre is in fact the real number line. These are the actual axes of that graph:

Perhaps surprisingly, zero is near one side of the cardioid (heart shape) whereas intuition would suggest it was at the bottom of Seahorse Valley where the circle and cardioid meet. It can be seen from this graph that the set is based on some kind of calculation involving real numbers, but what about the vertical axis?

The vertical axis represents the so-called imaginary numbers. These are numbers based on a concept which originally arose when it was realised that the square root of minus one seemed to be impossible. Since signs cancel out in multiplication, -1 x -1 is 1, so it clearly isn’t the real number one, and the only option appears to be to invent a second axis and think of numbers as existing on a plane as coördinates. These are known as complex numbers. They have both a real and an imaginary part. The word “imaginary” is used for want of a better term, as in fact these numbers are just as real as “real” numbers. There are also hypercomplex numbers such as quaternions and octonions which are a generalisation of this idea from the plane to space and hyperspace. On the whole, all of these numbers can be added, subtracted and the like, but the operations concerned don’t always have the same properties as those on real numbers. For instance, real number multiplication is commutative: 4 x 5 = 5 x 4. Octonion mutliplication is not, and this is crucial because for reasons I won’t go into here, it leaves the possibility that there is an omniscient observer open – it prevents Bell’s Theorem from being a proof of atheism.

The formula used to generate the Mandelbrot Set is quite simple, but before I get to that I’m briefly going to wallow in complex numbers for a bit. Complex numbers are combinations of real and imaginary numbers, more specifically the sum of a real number and the product of a real and complex number. They are useful, and here are two examples:

AC current is generated by a rotary dynamo and its voltage and current therefore vary as a sine wave – that’s the dynamo spinning. Capacitors and inductors alter this variation. Capacitors delay current, for example. The impedance, which is the opposition of a circuit to a current when voltage is applied. This can be calculated using complex numbers and power consumption can be reduced by doing these calculations to design efficient AC circuits.

An object with a rest mass can never move at the velocity of light because it would have infinite mass. However, if mass can be validly expressed by a complex number, this would not be a problem and therefore if tachyons – particles which only move faster than light – exist, they would have to have complex mass.

Hence complex numbers do have real world applications. They’re not only pure mathematics.

The Mandelbrot Set can be plotted on a plane without explicitly using complex numbers. There are programming languages, such as FORTRAN, which have complex numbers as a data type just as they do real and integer numbers, and in FORTH and other threaded interpretative languages there are ways of defining words which can perform operations on complex numbers as two values on the stack, but this isn’t necessary to plot the Mandelbrot Set, although it is effectively what’s happening when it’s done. The formula for the set is z_n+1 = z_n² + c, where z and c are a complex numbers and z is an integer. On a computer screen, each pixel is used as an example of c. This is a QBASIC program to generate the Mandelbrot Set in full:

DECLARE SUB InitPalette ()
DECLARE SUB DrawFractal ()
DECLARE SUB SaveScreen (filename AS STRING, w AS INTEGER, h AS INTEGER)

SCREEN 13
CLS
LINE (0, 0)-(320, 200), 15, BF

CONST w = 320
CONST h = 200

CALL InitPalette
CALL DrawFractal
CALL SaveScreen("MANDELBR.RAW", w, h)

REM This is fractal rendering code, the other functions are to make it look nicer
SUB DrawFractal
    DIM sx AS SINGLE
    DIM xy AS SINGLE
    DIM x AS SINGLE
    DIM y AS SINGLE
    DIM x2 AS SINGLE
    DIM y2 AS SINGLE

    DIM p AS INTEGER
    DIM r AS INTEGER
    DIM g AS INTEGER
    DIM B AS INTEGER

    CONST maxi = 100
    CONST colours = 256

    FOR py = 0 TO h - 1
        REM scale Y to -1:+1
        sy = (py / h) * 2! - 1
        FOR px = 0 TO w - 1
            REM scale x to -2.5:1
            sx = (px / w) * 3.5 - 2.5
            vy = 0
            vx = 0
            i = 0
            x = 0
            y = 0
            x2 = 0
            y2 = 0
            WHILE (x2 + y2 < 4) AND (i < maxi)
                xt = x2 - y2 + sx
                y = 2 * x * y + sy
                x = xt
                x2 = x * x
                y2 = y * y
                i = i + 1
            WEND
            c = i / maxi * colours
            PSET (px, py), c
        NEXT
    NEXT
END SUB


REM Changes the palette as the default is not pretty
SUB InitPalette
    DIM red AS LONG
    DIM green AS LONG
    DIM blue AS LONG
    DIM colour AS LONG

    FOR i = 0 TO 63
        blue = i
        green = i / 2
        red = i / 3
        colour = blue * 65536 + green * 256 + red
        PALETTE i, colour
        PALETTE i + 128, colour
    NEXT i
    FOR i = 63 TO 0 STEP -1
        blue = i
        green = i / 2
        red = i / 3
        colour = blue * 65536 + green * 256 + red
        PALETTE i + 64, colour
        PALETTE i + 192, colour
    NEXT i
END SUB

SUB SaveScreen (filename AS STRING, w AS INTEGER, h AS INTEGER)
    OPEN filename FOR OUTPUT AS #1
    FOR y = 0 TO h - 1
        FOR x = 0 TO w - 1
            PRINT #1, CHR$(POINT(x, y));
        NEXT x
    NEXT y
    CLOSE #1
END SUB

This is in QBASIC, Microsoft’s BASIC for DOS. It’s possible to zoom in by changing the ranges of py and sy in the control loops, and also their step value. The above code is both fancy and not optimised.

The first time I plotted the set, it was on an Acorn Electron. This was an eight-bit computer based on BBC Micro architecture, and was notably cut down from the glories of its predecessor. One of the ways in which this was done was by providing only four dynamic RAM chips and using them to store bytes in two halves, which meant that every time the CPU interacted with memory the clock speed was effectively halved – it had to fetch or store one byte in two cycles. It was possible to speed the computer up somewhat by fooling the video hardware into thinking it was using a text mode, because this skipped two scan lines per line of text and there was no need for the CPU to halt when the video hardware accessed video RAM, but this spoilt the display. The video modes I used to display the set were MODE 2 and MODE 0. The first of these is an eight-colour (supposèdly sixteen but half are merely flashing versions of the others) 256 x 160 display, and has the benefit of being colourful while not being very detailed. The other is 640 x 256 in two colours, allowing a lot of detail and to be honest I prefer it. Doing it on a BBC Micro would, for the reasons just mentioned, be faster than on an Electron, but that’s what I had. However, it’s dead easy to speed it up by only calculating the top or bottom half and mirroring it on the other half of the display. I think it took about eight hours to do the whole set.

I also did Seahorse Valley, which doesn’t benefit from any kind of reflection hack, and it took twenty hours. Not having a printer, I recorded it on a VHS cassette, which I still have somewhere. There are ways of speeding it up a lot, such as writing it in machine code and using fixed point arithmetic rather than floating point, and not bothering to calculate large blank bits of the picture, but I didn’t do those. Around the same time as I was doing this, a DOS program called FRACTINT was developed which managed to do it using only integer arithmetic and was therefore much faster.

The Mandelbrot Set is among the most complex objects in the Universe, or rather the Multiverse. It’s often claimed that the human brain is this, but in fact this object is far more complex because no matter how far you zoom in, there’s always more. It can also be extended into four dimensions because each point on it can be used to generate a Julia Set, such as this:

This can be thought of as a two-dimensional cross-section of an analogue of the Mandelbrot Set, as it varies continuously according to the location of the point used. One perhaps surprising fact about the set is that it took about fifteen years to prove that it was actually a fractal, over the period when it was all the rage and everyone was calling it one, when in fact it wasn’t known to be one. Also around this time it was proven to be as complex as it possibly could be.

The Mandelbrot Set itself does not reflect anything in the physical world, or rather the formula used to generate it seems not to have any practical application, but there is another set which is very similar and does represent something real, in the area of magnetism. This formula:

is quite reminiscent of the Mandelbrot Set’s, and describes what happens inside a magnetic material when it’s heated to the point where it’s completely demagnetised. A cold magnet is magnetic all the way through. All of its constituent parts which are individually magnetic are lined up. As it’s heated, the random movement of the atoms begins to dislodge the alignments in apparently random places, but they can in fact be predicted using this formula. If you plot this in the same way as the Mandelbrot Set, you get something like this:

From here. Will be removed on request.

There’s a blog mentioning this here.

Because of the visual effectiveness of fractal and this kind of imaging, it was suggested at some point that the nature of mathematical discovery had changed, because it was now possible to visualise much of what woul previously have seemed highly obscure. This has been seen as both bad and good. It’s good in that it makes maths more accessible and appealing, but it may also lead towards a bias towards the kind of maths that can do this kind of thing.

Finally, it occurs to me that my metaphor for consciousness being a property of matter like magnetism could be extended meaningfully to model a dying brain. What if the way consciousness works involves a whole, fully awake and living brain as one of the stable states, the “dark bits” as it were on the Mandelbrot Set, but that on falling asleep, having a seizure, being starved of oxygen or under the influence of drugs it fragments consciousness physically into little areas throughout the brain which are individually conscious but not unified? And what if this could be modelled mathematically? I don’t know where I’m going with this but it sounds promising.