These calculations are much more complicated and require an enormous amount of computer time, because a very large number of [points] (more than 5000) must be taken to achieve convergence… [our] calculations are thus very fast and convenient, and indeed we were able to perform them on a PC microcomputer.

– some scientist in 1988 (emphasis mine)

http://thesummerofmark.tumblr.com/post/93894004721/thesummerofmark-my-favorite-proof-that-the

thesummerofmark:

thesummerofmark:

my favorite proof that the fundamental group of the circle is the additive group of integers is the one where you imagine something going around in a circle, in either direction, arbitrarily many times

Err, sorry—I definitely meant higher homotopy groups, not higher spheres. I agree that the fundamental group for higher spheres isn’t so challenging, especially if you’re practiced at it.

Negative radiation forces act opposite to the direction of propagation, or net momentum, of a beam but have previously been challenging to definitively demonstrate. We report an experimental acoustic tractor beam generated by an ultrasonic array operating on macroscopic targets (>1 cm) to demonstrate the negative radiation forces and to map out regimes over which they dominate, which we compare to simulations. The result and the geometrically simple configuration show that the effect is due to nonconservative forces, produced by redirection of a momentum flux from the angled sides of a target and not by conservative forces from a potential energy gradient. Use of a simple acoustic setup provides an easily understood illustration of the negative radiation pressure concept for tractor beams and demonstrates continuous attraction towards the source, against a net momentum flux in the system.

– Abstract: “Acoustic Tractor Beam”, PRL 112, 174302 (2014)

This “function” has a superficial similarity to ‘unsafePerformIO’ but it is in fact a malevolent agent of chaos. It unpicks the seams of reality (and the ‘IO’ monad) so that the normal rules no longer apply. It lulls you into thinking it is reasonable, but when you are not looking it stabs you in the back and aliases all of your mutable buffers. The carcass of many a seasoned Haskell programmer lie strewn at its feet…

Do not talk about “safe”! You do not know what is safe! Yield not to its blasphemous call! Flee traveller! Flee or you will be corrupted and devoured! …

If you think you know what you are doing, use ‘unsafePerformIO’. If you are sure you know what you are doing, use ‘unsafeDupablePerformIO’. If you enjoy sharing an address space with a malevolent agent of chaos, try ‘accursedUnutterablePerformIO’.

– from the source code for ‘accursedUnutterablePerformIO’ and ‘inlinePerformIO’, from Data.ByteString.Internal

The brain is wider than the sky,
For, put them side by side,
The one the other will include
With ease, and you beside.

The brain is deeper than the sea,
For, hold them, blue to blue,
The one the other will absorb,
As sponges, buckets do.

The brain is just the weight of God,
For, lift them, pound for pound,
And they will differ, if they do,
As syllable from sound.

thesummerofmark:

i think there is too much emphasis on visual intuition in geometry and not enough emphasis on haptic intuition, especially when it comes to higher-dimensional stuff

remember, you can put your hands on opposite sides of a shape, but you can’t _see_ opposite sides of a shape

For the same reason that we no longer use a certain king’s feet to measure distance, we use natural units in which the speed of light c and the Dirac symbol ħ are both set equal to 1.

A Nice Cup of Tea

By George Orwell

Evening Standard, 12 January 1946.

If you look up ‘tea’ in the first cookery book that comes to hand you will probably find that it is unmentioned; or at most you will find a few lines of sketchy instructions which give no ruling on several of the most important points. This is curious, not only because tea is one of the main stays of civilization in this country, as well as in Eire, Australia and New Zealand, but because the best manner of making it is the subject of violent disputes. When I look through my own recipe for the perfect cup of tea, I find no fewer than eleven outstanding points. On perhaps two of them there would be pretty general agreement, but at least four others are acutely controversial. Here are my own eleven rules, every one of which I regard as golden:

First of all, one should use Indian or Ceylonese tea. China tea has virtues which are not to be despised nowadays — it is economical, and one can drink it without milk — but there is not much stimulation in it. One does not feel wiser, braver or more optimistic after drinking it. Anyone who has used that comforting phrase ‘a nice cup of tea’ invariably means Indian tea.

Secondly, tea should be made in small quantities — that is, in a teapot. Tea out of an urn is always tasteless, while army tea, made in a cauldron, tastes of grease and whitewash. The teapot should be made of china or earthenware. Silver or Britanniaware teapots produce inferior tea and enamel pots are worse; though curiously enough a pewter teapot (a rarity nowadays) is not so bad.

Thirdly, the pot should be warmed beforehand. This is better done by placing it on the hob than by the usual method of swilling it out with hot water.

Fourthly, the tea should be strong. For a pot holding a quart, if you are going to fill it nearly to the brim, six heaped teaspoons would be about right. In a time of rationing, this is not an idea that can be realized on every day of the week, but I maintain that one strong cup of tea is better than twenty weak ones. All true tea lovers not only like their tea strong, but like it a little stronger with each year that passes — a fact which is recognized in the extra ration issued to old-age pensioners.

Fifthly, the tea should be put straight into the pot. No strainers, muslin bags or other devices to imprison the tea. In some countries teapots are fitted with little dangling baskets under the spout to catch the stray leaves, which are supposed to be harmful. Actually one can swallow tea-leaves in considerable quantities without ill effect, and if the tea is not loose in the pot it never infuses properly.

Sixthly, one should take the teapot to the kettle and not the other way about. The water should be actually boiling at the moment of impact, which means that one should keep it on the flame while one pours. Some people add that one should only use water that has been freshly brought to the boil, but I have never noticed that it makes any difference.

Seventhly, after making the tea, one should stir it, or better, give the pot a good shake, afterwards allowing the leaves to settle.

Eighthly, one should drink out of a good breakfast cup — that is, the cylindrical type of cup, not the flat, shallow type. The breakfast cup holds more, and with the other kind one’s tea is always half cold before one has well started on it.

Ninthly, one should pour the cream off the milk before using it for tea. Milk that is too creamy always gives tea a sickly taste.

Tenthly, one should pour tea into the cup first. This is one of the most controversial points of all; indeed in every family in Britain there are probably two schools of thought on the subject. The milk-first school can bring forward some fairly strong arguments, but I maintain that my own argument is unanswerable. This is that, by putting the tea in first and stirring as one pours, one can exactly regulate the amount of milk whereas one is liable to put in too much milk if one does it the other way round.

Lastly, tea — unless one is drinking it in the Russian style — should be drunk without sugar. I know very well that I am in a minority here. But still, how can you call yourself a true tealover if you destroy the flavour of your tea by putting sugar in it? It would be equally reasonable to put in pepper or salt. Tea is meant to be bitter, just as beer is meant to be bitter. If you sweeten it, you are no longer tasting the tea, you are merely tasting the sugar; you could make a very similar drink by dissolving sugar in plain hot water.

Some people would answer that they don’t like tea in itself, that they only drink it in order to be warmed and stimulated, and they need sugar to take the taste away. To those misguided people I would say: Try drinking tea without sugar for, say, a fortnight and it is very unlikely that you will ever want to ruin your tea by sweetening it again. These are not the only controversial points to arise in connexion with tea drinking, but they are sufficient to show how subtilized the whole business has become. There is also the mysterious social etiquette surrounding the teapot (why is it considered vulgar to drink out of your saucer, for instance?) and much might be written about the subsidiary uses of tealeaves, such as telling fortunes, predicting the arrival of visitors, feeding rabbits, healing burns and sweeping the carpet. It is worth paying attention to such details as warming the pot and using water that is really boiling, so as to make quite sure of wringing out of one’s ration the twenty good, strong cups of that two ounces, properly handled, ought to represent.