Nightwish

I'm (kind of) sorry to say I'm acquiring a reputation among my friends for liking weird music (though I don't find it weird). I'm not sure what it means, but I keep on running into music and liking it, and finding it's Finnish. As I said elsewhere - see Finnish folk roots - the Finnish music scene appears to have a strong fusion element, a strong crossover between genres such as folk and pop and metal, that would be marginalised and viewed as a curiosity elsewhere. Anyhow, check our Nightwish, a Finnish metal band (see their official YouTube channel) who do excellent metal / folk / operatic /rock fusion. Examples: Over the hills and far away; Bless the Child, Ocean Soul, this rather brilliant cover of Phantom of the Opera, and Nemo.

PS: this is altogether rather interesting. I'm going to have to investigate symphonic metal.

Friday Automata Blogging: Hanging Around

Friday Automata Blogging: The Birds

Gone astray ...

A sphere coloured by surface area in a ratio 1:φ. The red upper section subtends at the centre of the sphere Professor Greg Parker's Golden Solid Angle.

Oh, dear. Further to the previous post, I notice that Professor Greg Parker writes in Some more on the Golden Solid angle (continuing the topic of his request for natural instances of a solid angle of 1.52786*π steradians):

Some people in trying to help with a reply have gone astray with both the mathematics involved (which aren’t that complex) and the concept.

I assume this means me, as I've been the only one to comment so far. No, I haven't gone astray. My analysis produced exactly the same result as Greg's for gamma, his Golden Solid Angle (γ = 1.52786*π = 4.799913751 steradians) so nothing is wrong with that side of things.

What I have done, that he seems to dislike, is to analyse it in terms of a practical working way to identify the magic angle he's looking for. No-one, not even a specialist, can be expected to look at an object and say, "Aha, that's a solid angle of 1.52786*π steradians." His definition is perfectly accurate mathematically, but near-useless practically. We need a visualization for a rough guide, which I've provided (above), and a simple way to ascertain more accurately whether an object has that solid angle.

Here's an example. Do the white end caps of the pool ball (left) subtend a Golden Solid Angle at its centre?



There's no way to reach inside and apply some hypothetical ghostly solid-angle protractor, but what we can do is look at the cross-section (right - not too close, to avoid perspective distortion) find the centre O and the angle θ subtended by the edge of the end cap.

Then the solid angle subtended by the cap = 2*π*(1-cos(θ))



We can see it's about 55 degrees, so the solid angle is

2*π*(1-cos(55*π/180)) = 2.679298269 steradians

So, no, our pool ball end cap is too small for γ which is 4.799913751 steradians. As I said in the previous post, we're looking for an object where θ is about 76.345 degrees. The same kind of analysis would apply if we were looking at something other than a sphere, such as a cone-shaped plant or animal structure. There's no way to measure solid angle directly; you need to measure θ on a cross-section, then calculate the solid angle using 2*π*(1-cos(θ)).

The angle at the blunt end of some species of cone shell, incidentally, looks as if it might be near the required angle.

PS: Sorry about the problem with the comments; I switched them off while using this site to prototype another. Comments are now enabled (and I actually publish mine).

Update: Greg has posted an update More on the Golden Solid angle still disputing my view that's there's not likely to be significance in dividing a sphere's surface area in the ratio 1:φ. He writes:

Unfortunately this is quite incorrect! Go down one dimension to the Golden (planar) angle of roughly 137.5 degrees and you will find this angle appearing time and time again in the subject area of Phyllotaxis

Yeah, I know. But this is an essentially fallacious argument: that because something works in 2D, it will if generalised to a different dimension. It won't necessarily: an example in this very area is the Golden Rectangle, which has the nice property that if you chop a square off the end, you get a smaller Golden Rectangle. This property doesn't generalise to 3D; there is no cuboid where you can chop a cube off the end and get a smaller object geometrically similar to the original. Or another geometrical example: equilateral triangles tessellate on a plane thus; but regular tetrahedra can't tessellate 3D space.

There's no reason to assume a geometric property that depends on the specific properties of 2D packing will work in 3D.

Discovered or invented?

A posting at Felix Grant's weblog The Growlery - Oh, that this too too solid angle... - just made me realise I've been astonishingly remiss. A considerable time ago, Simon & Schuster sent me a copy to review of Mario Livio's Is God A Mathematician? Ouch - it arrived right in the middle of panicking about my tax return and got buried under paperwork .. and it just resurfaced. I'll write about it, I promise! But first...

Felix is asking for response to Professor Greg Parker's The Golden Solid Angle – first written for publication 14th June 2007. The idea is this: imagine a sphere with a circular spot on it, like the 8-ball in pool. Greg - I'll use that for brevity as Felix does - proposes such a sphere where the spot has an area proportional to 1, and the remainder of the surface an area proportional to φ (phi aka the golden ratio) whose value is (1+√)/2 ... about 1.618034). This spot would subtend a solid angle Greg calls The Golden Solid Angle. He writes:

So I wrote a paper on “The Golden Solid Angle” for the Mathematical Gazette, which was in fact turned down as “although the result was new, just having a new result is not necessarily having something worthy of publication” – well that’s a new one for me! So wishing to stake my claim as the discoverer of the Golden Solid Angle (sent to the Mathematical Gazette on Thursday 14th June 2007) here’s the thing explained for the first time below.

This very much impinges on the subject of Is God A Mathematician?: Mario Livio - who has also written a book on the golden ratio - asks if mathematics is "discovered" or "invented". To cut to the chase - there's a review by Marianne Freiberger in Plus magazine - a bit of both.

π is a classic example of a "discovered" number. Apart from being the ratio of a circle's radius to its diameter, it's ubiquitous in mathematics, cropping up as the result of integrals, statistics, various forms of numerical series, and so on, that appear to have nothing to do with it. φ, on the other hand, Livio considers to be "invented". This is perhaps a harsh assessment, in that it does crop up in natural forms where the classic Fibonacci sequence generates it as a ratio. But it doesn't surface repeatedly in mathematics to the sheer extent of π. As the Plus magazine review puts it:

Much like the umbrella was invented in England and not the Sahara, so was the concept of the golden ratio invented by the Greeks, and not the Indians or Chinese. The Greeks' preoccupation with geometry brought them into frequent contact with this ratio, and so they needed a name for it — beyond that, there's nothing universal about this particular object.

There is, in addition, a long history of (having decided φ to be a significant number) slotting it into contexts that wouldn't naturally or mathematically produce it, then ascribing significance to them - e.g. a circle with diameter φ is the Golden Circle - and that is pretty unambiguously invention. And that brings me back to Greg's concept of the Golden Solid Angle (which divides the surface area of a sphere in the ratio 1:φ). I think it would be a "discovery" if it had surfaced unexpectedly from some analysis; but as it stands, it seems to be the result of an arbitrary choice to put φ in as a value - an invented situation that could just as easily have been a ratio of 1:π or 1:3 or 1:e. With definitely no insult intended to Greg's explorations of interesting mathematical relations, I can understand the Mathematical Gazette's "So what?" response.

Unless ... meaningful "discovered" occurrences of this Golden Solid Angle can be found. If you see a sighting in the wild or know a mathematical process (such as an optimisation) that makes an object with a Golden Solid Angle γ = 4.799926453, let Greg know.

A sphere whose surface is divided in that proportion will look like this:



Derivation, for those interested



It's easier to do the analysis in cross-section. We'll take a sphere of radius = 1 for convenience.

A sphere's surface area = 4*π*r^2 = 4*π in our case.

The surface area of the spherical cap = 2*π*(1-cos(θ))

We want the sphere's surface area divided in the proportion cap:remainder = 1:φ
which means the fractional areas of the cap and the rest are 1/(1+φ) and φ/(1+φ)
or in actual areas (4*π)/(1+φ) and (4*π*φ)/(1+φ)

Now we can equate the two expressions for area to get θ. For the cap:

2*π*(1-cos(θ)) = (4*π)/(1+φ)
1-cos(θ) = 2/(1+φ)
cos(θ) = 1 - 2/(1+φ) = (φ-1)/(φ+1)

θ = ACOS((φ-1)/(φ+1))

Plugging in φ = (1+sqrt(5))/2 we get

θ = ACOS(sqrt(5)-2) = apprx. 1.332478864 radians = apprx. 76.34541519 degrees

Just to confirm the result: this will give a surface area for the cap (and solid angle for the whole cap) of:

2*π*(1-COS(1.332478864)) = 4.799926453 = 1.527864042*π

which corresponds with Greg's result at Some more on the Golden Solid angle obtained by directly solving for surface area ratio (4*π-γ)/γ = 4*π/(4*π–γ)
which gives γ = 4.799926453

(While it's a little longer, I actually prefer my analysis in terms of θ because angle seen in cross-section is a parameter more readily understood than solid angle).

Unreal instruments

Clare just got an e-mail that Mr Know-it-all instantly spotted as a hoax:

Read this first, then watch.

AMAZING!

Turn your sound on for this.

This is almost unbelievable. See how all of the balls wind up in catcher cones.

This incredible machine was built as a collaborative effort between the Robert M. Trammell Music Conservatory and the Sharon Wick School of Engineering at the University of Iowa ... Amazingly, 97% of the machines components came from John Deere Industries and Irrigation Equipment of Bancroft , Iowa ...Yes, farm equipment!

It took the team a combined 13,029 hours of set-up, alignment, calibration, and tuning before filming this video but as you can see it was WELL worth the effort. It is now on display in the Matthew Gerhard Alumni Hall at the University and is already slated to be donated to the Smithsonian.

The accompanying clip was in the fact the above, Pipe Dream, minus the onscreen credits, from the music animation specialists Animusic. I've expounded a bit more on their lovely works at JSBlog: see Self-playing harps.

Maybe I've watched too much computer animation, but I can't see how anyone could think this to be real for more than a moment. Nor did I realise how ubiquitous the e-mail is: enough to be mentioned on a number of debunking sites such as Snopes. Needless to say, the Robert M. Trammell Music Conservatory and the Sharon Wick School of Engineering at the University of Iowa do not exist. The discussion at The Blog of Phyz - Fooling our elders... - is enlightening if depressing; I know well the syndrome described, where the person who spots a hoax becomes cast as the bad guy:

The person who sends the hoax is regarded as a happy-go-lucky victim with a positive outlook on life, but the person who responds with the truth is regarded as a curmudgeonly killjoy.

The alternative, I suppose, is the recipient admitting being fooled. In this case, however, the hoax is doing down a remarkable piece of work, and by removing the credit may even count as video piracy. As the Hoaxslayer entry says:

There is no need to malign this fantastic animation by tacking on a foolish and totally fictitious cover story. Such clever work speaks for itself and needs no embellishment. Moreover, the real creators of the animation deserve credit for their genius. If you receive this email forward, please let the sender know the true origin of the "farm machine music" video.

Crab collars

I just had to preserve this, recently posted on Wikipedia and not unnaturally speedy-deleted as a hoax.

Crab collars

From Wikipedia, the free encyclopedia

Overview

Collars have been recommended for people who have pet crabs of the following varieties:

• Hermit (All Paguroidea varieties)
• Halloween (Geocarcinus quadratus)
• Japanese spider crabs (Macrocheira kaempferi)

Purpose

Many people don't consider the importance of collars for pet crabs. Some key reasons collar users are listed below:

1. Identification
2. Protection
3. Listed Medical Allergies
4. Contact Number / Return Address
5. Bling

• Users have found peace of mind especially when having to ask their pets to be minded with the added support that a collar offers.

Fitting

• Carefully clean the pet before attaching to help provide a close bond with the collar.
• Ensure that the collar is attached firmly to the to the crab's largest Merus as to stop the pet from being able to detach the collar with a cheliped.
• Do Not attach a collar to a main shell for Hermit varieties as the pet can simply remove itself from the housing. Special collars made for these varieties are to be attached to the Cephalothorax
• Tighten the collar just enough so that it cannot move or twist.
• Choose a colour that compliments your crab, the delicate social structure of crabs can greatly be affected due to favourtism and rejection.

Usage

• A fitted collar can provide a great deal of fun outside of the confides of your tank or aquarium. Well exercised crabs are happy crabs
• When swimming with your pet, ensure you take an extended lead to ensure maximum freedom. As crabs are predominately land crawlers they may stress do to prolonged or restricted underwater activity.
• When requiring your pet to be cared for temporarily by a friend or relative, ensure your collar is up to date with details.