Date
created 21. Mayıs 2005
Source
Own work
Author
Manuel Heinrich Emha
The other day a bloke turned up at our front door with what looked like a scythe wrapped in brown paper. It was easy to distinguish him from the Grim Reaper because he wasn’t wearing a robe and had a lot more flesh on his bones than one might expect. It was in fact my dad’s hoist. Rather taken aback that there didn’t seem to be a base, I asked him about the contents of his truck and he replied that there was a second piece whose shape he had difficulty in describing. I then realised, suddenly, that this was an opportunity to use a word which practically nobody knew, although it would not be an aid in communication. He was attempting to describe a rectangular balbis!
For some reason, although it’s a common shape the word is almost never used, and as far as I know has only been written about once in geometry. There are two types of balbides (that’s the plural). One is in the shape of a capital H, which is the “common” balbis, and the other is like a rectangle with one side missing, the rectangular balbis. They might at first be considered to be different shapes, but depending on the form of geometry used, they may or may not be the same. Remarkably, only one mathematician ever seems to have studied and written about this shape and it’s generally dismissed or ignored, but like other mathematical figures it has properties of its own which seem just as significant as others.
In geometry, a literal line is infinite in length. It’s a line segment which isn’t. In Euclidean geometry, parallel lines never meet, but it emerged in the nineteenth century that Euclidean geometry doesn’t apply to our Universe and in fact parallel lines do meet at a distance of many gigaparsecs. Now consider both types of balbis. In Euclidean space, a rectangular balbis can be thought of as a triangle with two right angles whose apex is at infinity. It’s a limiting case of a triangle which can only exist in Euclidean space. In space as it actually is, or seems to be, there can be no absolute rectangular balbis because space ultimately curves round on itself, so the only possible type of balbis is the H-shaped version, whose sides come back round the Universe to meet at their starting point. However, there can be a shape very close indeed to a rectangular balbis in the form of an isosceles triangle which is very nearly a rectangular balbis but in fact has an apex higher than the distance between the base and the edge of the observable Universe. It’s hard to imagine any practical purpose for this, and also hard to imagine any observable property this triangle might have that distinguishes it from a rectangular balbis.
As I’ve mentioned, there has so far as I can tell only ever been one mathematician who has studied or written about balbides. His name was the Reverend P H Francis, he held a Masters from Cambridge in Mathematics and he is remarkable among the people I might describe as fringe thinkers for being a proper scientist, in a way. His work focussed on three areas. Two of these were the analysis of games in a non-game theory kind of way and the mathematics of infinity, particularly in connection with balbides. His understanding of games is that they are an outgrowth of the human instinct for hunting and are all connected to the idea of aiming for a target, which it occurs to me might be interesting from a theological viewpoint in view of the fact that the generic Hebrew word for sin, עבירה, literally means missing the target, and I note the guy was a vicar. His mathematics presumably makes sense and is up to a certain academic standard. From what I’ve read of his work, it made sense to my twelve-year old mind, bearing in mind that I have no particular aptitude in that field. He certainly seems to have been a respectable mathematician, if somewhat isolated because of his idiosyncratic interests. His third field is where it gets weird. He believed that the nature of infinity offered a solution to Olber’s Paradox, and his solution was utterly bizarre. Before I get to that, I should go into what Olbers’ Paradox is.
Olbers’ Paradox can be summed up by the question, why is the sky black? That is, if we live in an infinite Universe (and we probably don’t but bear with me because it’s still odd) which is homogenous, i.e. there are stars and galaxies in all directions, when we look up at the night sky it ought to be blue-white at a colour temperature of around 40 000°C, and in fact the temperature of most of the Universe should be at around that level. This is because, space being infinite in this view, any straight line from any point should intersect with the surface of one of the hottest possible stars at some point, so the entire “surface” of the sky should be that bright and that hot. But it isn’t. Why?
Some of the assumptions on which this is based are now outdated. In the early twentieth century, the Steady State theory held that space was infinite and eternal, and this view was remarkably well-supported by the evidence available at the time. Once it was realised that space was constantly expanding it became clear that another solution to the paradox was that over sufficient distances, space was expanding faster than light and therefore light would never reach us. Incidentally, the issue is also present to some extent in a finite but static Universe, since if space is curved light is going to travel a lot. This also means that if the Universe was contracting rather than expanding, it would also heat up for the same reason and life would become impossible – every point would catch up with all the blue-shifted light. For this reason, it’s been stated that “I think therefore I am, and therefore the Universe is expanding”, which is probably the second thing Deep Thought realised.
P H Francis came up with a different solution to Olbers’ Paradox, apparently partly based on his views of infinity. I’m straining my memory here, but I seem to recall that he believed that the real number line was in fact a loop, reaching positive infinity before reversing its sign and becoming negative infinity, so for him a truly infinite number was both positive and negative. Consequently, for him too a balbis would be an example of a shape which meets itself in the middle, but for different reasons. It also seems to mean for him that at infinity, light reverses its direction, meaning that space is not exactly limited but reflects like a mirror. This next bit I have to admit I find utterly baffling. Francis also believed that infinity could be at any distance, possibly because the Universe is an irregular apeirohedron. This is going to need some explaining, and I may be wrong. An apeirogon is a polygon with countably infinite sides. An apeirohedron is the three-dimensional version, and is not the same as a sphere or spheroid because these have uncountably infinite faces. Instead, an irregular apeirohedron would consist of an infinite number of convex and concave polygons, and I’m guessing that this is what Francis has in mind. Because infinity is in a sense undefinable, it means that there are many possible values. Note that I don’t believe this is true, but in making that statement, if my guess is correct, I am contradicting a qualified mathematician who presumably knew what he was talking about.
And there’s more. Francis claimed further that the only star in the Universe is the Sun itself. What we see as stars in the night sky are in fact reflections of the Sun at various distances as its light “bounces off infinity”. This is his solution to Olbers’ Paradox. Moreover, the Sun is not hot. He believed that “the popular notion that the Sun is on fire is rubbish, and merely a hoary superstition on a par with a belief in a flat earth, an Earth resting on the back of a tortoise or an elephant, or a sun revolving around a stationary earth.” Of course people don’t literally believe that the Sun is on fire, but his target is more the idea that the Sun is hot at all. He gives several reasons for supposing the Sun is not a hot ball of gas. Firstly, the Sun is roughly spherical, and if it were a ball of gas or plasma, it would not have a smooth surface. Secondly, space is a vacuum, and heat cannot travel through a vacuum, hence thermos flasks. Thirdly, heat can be generated by cold objects. An electric fire, for example, is hot, but the generator which provides the current to be converted to heat needn’t be. He believed instead that the Sun is an electrically-charged light source whose electricity warms Earth’s atmosphere, and therefore the surface, and one piece of evidence for this is that at higher altitudes it’s colder, because the molecules of the atmosphere are further apart and therefore the heating effect is weaker. Earth is also reflective, and this prevents the radiation of heat into space because, and I may not be following this exactly, silvered surfaces do not radiate heat but prevent it from being radiated.
All of this is very odd. Whereas I don’t believe it for a second, that isn’t really what’s odd about it. The Rev. Francis reached his own conclusions which were well-founded, as he saw it, in mathematics, and there are clear links between his interests. A rectangular balbis is a goal mouth that never ends and he was interested in games. He saw games as centred on the idea of achieving a target, which he also saw as an evolutionary imperative connected with survival via hunting. At the same time, he was a vicar so he may have had similar views on Christianity as a useful strategy for playing the game of life, which involves meeting a moral target. Then there’s the issue of what happens if a balbis goes on forever and reverses into itself – infinity. Then there’s Olbers’ Paradox, which he seems to have solved by using the concept of infinity as he saw it and the reflection of light in an undefined distance. In fact the only bit of his thought which seems not to be part of this coherent whole is the temperate Sun. Even so, it has an internal consistency to it. What’s odd about it is him. He’s a qualified mathematician who managed nonetheless to draw the same kind of conclusions about the nature of reality as might be reached by a Flat Earther, someone who believes we are within a hollow earth, that Venus was a comet, that aliens visited humans in prehistoric times or that there is phantom time and the dark ages never happened. Some of the people involved in these claims are educated and intelligent to be sure. For instance, Velikovsky (the Venus guy) was a psychiatrist, Illig (phantom time) seems to have been his acolyte, Von Däniken was a hotel manager who, however, probably originally got his ideas from Carl Sagan and doesn’t appear to be sincere, and Flat Earthers are a whole plethora of people who, however, tend not to be scientifically trained. P H Francis is not like this. The profession followed by Velikovsky is potentially scientific but he seems mainly to have been a Freudian psychoanalyst which is clearly not. But Francis was a respectable mathematician, although many of his works are self-published. How did it happen that he reached such heterodox conclusions about cosmology?
He isn’t, in fact, alone in this. Fred Hoyle also ended up drawing unexpected conclusions and sticking with them, although they were somewhat more in keeping with mainstream cosmology. Hoyle was the first person to hypothesise that heavy elements were formed by nuclear fusion in stars, something which Francis, incidentally, definitely would not be on board with. This is now accepted more or less universally. However, he rejected the Big Bang theory and stuck with the Steady State, claiming that the apparent red shift of receding objects was not caused by the expansion of space but the presence of microörganisms in the interstellar medium absorbing other wavelengths of light. Hoyle in fact coined the term “Big Bang Theory”, in 1949, and meant it to be pejorative. He believed that the Roman Catholic priest Georges Lemaître who originally came up with the idea that because space was expanding it must have originally been in a hot, dense state at the beginning of time to be akin to the cosmological argument for the existence of God, and to be honest I find it hard not to agree with him there. But those are my issues and not those of someone more versed in cosmology than I. Hoyle did, however, believe in the fine-tuning argument, as he held that the existence of the carbon atom in particular seemed suspicious in an arbitrary Universe.
One significant difference between Hoyle and Francis seems to be that Hoyle was inside his profession and Francis outside it. Consequently Hoyle is not perceived as a “crank” in the same way as someone like Velikovsky or Illig, whereas Francis probably is. After all, he did self-publish and doesn’t seem to have had academic peers. It’s also interesting that there are two priests here involved in cosmology, one an Anglican, the other a Roman Catholic, and I think this perhaps illustrates how cosmology itself, as James Muirden claimed, is not a purely scientific profession but attracts people who would seek non-scientific but nonetheless valid answers to ultimate questions about reality. It’s substantially about reputation and being linked to some kind of social network, and that isn’t just to say it’s an “old boy network”, although I think it is, but that we all need to bounce our ideas off people to remain sane. Nonetheless, the takeaway from this is that the Reverend P H Francis stands out among “cranks” by being so very heretical in spite of being scientifically and mathematically literate, and I think this makes him unique.
A Box Of Chocolates 