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.

Gauge Invariance

fuelbydreams replied to your post:semisemiclassical electrons What is gauge invariance?

Sometimes, the mathematical objects that are useful to think or talk about aren’t actually measurable in the real world. Instead, you have to apply some computation on them to get a “real” result.

For example, when you measure “voltage” you’re really measuring a difference in two electrical potential energies. You can’t really assign a meaningful value to either energy independently; only their difference is important.

Likewise, the main object of quantum mechanics—the wavefunction—lets you calculate a probability by taking its absolute value. Therefore, multiplying the wavefunction by a sign (or, more generally, a complex number on the unit circle) won’t change the outcome probability.

There’s a kind of generalized electrical potential energy called the vector potential. You can add certain quantities to it, as well, and it turns out that when you go to calculate the electric or magnetic fields whatever you added (or didn’t add) gets thrown out of the calculation.

All of these are examples of gauge invariance. Because you can freely modify those values I mentioned, they are in some sense “unphysical”; you couldn’t measure them in the real world and expect to get one consistent outcome.

In my earlier post, some scientists were talking about sources of magnetization and tried to break the total magnetization into two contributions: one contribution from electrons moving around their own centers of mass, and another contribution from the electrons moving through the material. But it turns out that these two contributions aren’t really gauge invariant: by freely changing some numbers, which we are allowed to do (you can always “pick a gauge” since it doesn’t effect the outcome), we can see the source of the magnetization shift between these two suggested contributions. Therefore, neither of these contributions are physical, because you couldn’t measure them individually and get a consistent answer (though their combination, and in fact a slightly different division of magnetization sources that the authors go on to discuss, is gauge invariant and can be determined in principle/IRL).

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With over 700 new functions—the single biggest jump in new functionality in the software’s entire history—Mathematica 10 is the first version of Mathematica based on the complete Wolfram Language. Integration with the Wolfram Cloud and access to the expanded Wolfram Knowledgebase open up many new possibilities for intelligent computation and deployment.

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– To: danielsmw@cmu.edu

semisemiclassical electrons

When discussing the magnetic properties of materials, one has to consider the orbital magnetization due to electron circulation in the material, which act very schematically as little current loops giving rise to magnetic fields.*

It turns out that if you try to think about this contribution semi-classically, you can separate the orbital magnetization into two parts: one gauge-invariant part which is due to some self-rotation of the electron wavepackets about their centers-of-mass, and then another part that’s due to the “itinerant currents” of the electrons moving in “current loops”.

You can make that separation, but the two parts aren’t gauge invariant. By changing the gauge, you can move angular momentum between self-rotation of the wavepackets and the wavepackets’ motion through the material.

The lesson being not to think about electrons semi-classically, I guess.

* Though, in most materials, the magnetization is due almost entirely to electron spin.

PS (mostly @TSoM): What does topos theory have to say about gauge invariance?

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