Reach for the Moon

And to dispel any ideas I had completely gone....



I caught this quite by accident when walking from the KFC to my Liverpool place. My NEW Liverpool place, courtesy of my stalker, DPH.

It was about 5 pm , GMT, on the last visit but one, about 100 yards from my front door.

I don't JUST take portraits!

See you in the Partial Differentials!

Potentially Fascinating

Concrete highways and railway lines. Ever noticed that when a car travels along the former, or a train along the latter, they sound just a little bit similar?

"Bee-dum Bee-dum Bee-dum..."

It is because concrete highways are made of slabs of concrete, and the railway lines, sections of rail.

The reason is simple - seasons! Both are cold in the winter, and warm in the summer.

Heat makes things expand, and if those structures were continuous, the concrete would crack, and the rails buckle.

So the engineers put bitumen between the concrete slabs, and air gaps between the rails, to take the strain, so to speak.

The sound

"Bee-dum Bee-dum Bee-dum..."

Is due to the Doppler effect, which is the same effect you hear as police sirens go past - the pitch sounds lower, but in this case, it is the front wheels versus the back wheels of the vehicle. It is more pronounced for cars as the two pairs are further apart, and the "dum" bit is distinctly behind.

So, when things heat up, they get larger.

They expand.

BUT we don't want to give you that! Since this blog is officially in second gear, there is a need for further thought - the question of WHY this happens.

It is really quite simple.

It is called the "Lennard-Jones 6-12 Potential".

It is an empirical formula (that is, it is based on observation rather than any theory) but it fits the data rather well.

It's proper form is:



Which can be reduced to:



The "r" is the distance between atoms next to each other, and "V" is the amount of energy they have when added together. "A" and "B" are just numbers.

The comparison between this equation and real life can be seen here:



...which is pretty good.

Back to the question: WHY do substances expand when heated?

It is a question of energy. If you heat something up, you give them energy, and if the thing is jiggling, it will jiggle more if it has more energy. There is a related phenomenon called Simple Harmonic Motion, which is illustrated here:



The only difference, for the purposes of this post, between SHM and the Lennard-Jones is that the former is symmetrical, the latter is not. If you twang the ruler further, it will jiggle further from the center, but will always have the same centre. The ruler stays and bounces back and forth across where it normally wants to be. As does the weight, the spring, and the pendulum. Graphically, the energy (or "potential") will look like this:



which is symmetrical, as can be seen by the vertical line. The line is vertical because the ruler, etc still goes back and forth around the same centre point.

Now take another look at the Lennard-Jones graph. It has a similar shape, but if you try to draw a "symmetry" line, it will bend off the vertical to the right. If the central point of the jiggling gets further away from zero, it gets larger, and what we see is that the two atoms actually increase their separation.

The English words for this are "the substance expands".

Told you it was simple.

However, what's the deal with water? If you cool water so that it freezes, what happens? Ice floats and glass bottles shatter BECAUSE - water expands on going from water to ice - the complete opposite!!

Watch this space.

Bang Out of Order

It does not stop there, though.

The "last post" considered only homogeneous linear differential equations (with constant coefficients) of the first order.

What about second order?

In order to explain it is necessary to recap just what a "differential" is.

The question arose because people wanted to find out how steep hills are.

How steep are they? Well, we say 1 in 10 or 1 in 4, etc, meaning that for every meter we move up vertically, we have to travel 10 or 4 meters along the road to get there. Like this:



In mathematics, we have to be more precise and say not "along the road" but " horizontally".

This is the basis of all gradients. In the diagram below, k is the gradient of each line:



BUT being mathematics, straight lines were too restrictive, as most graphs are actually curved. Enter the CALCULUS.

Calculus gave us the way to find the gradient at any point on a curve, by measuring the gradient of the "tangent" at that point:

The procedure was quite simple. Take one of the simplest curves, a parabola, whose curve is described by y = x^2



Notice how the tangents at x = 1 and x = 2 have gradients of 2 and 4 respectively. Almost like the gradients are twice the x-value of the points.

In fact, that is exactly what is happening. For every point x on y = x^2, the gradient is 2x.

By analogy, for:

y = x^3, the gradient of every point is 3x^2

y = x^4, the gradient of every point is 4x^3.

....there is obviously a pattern here.

The pattern is this:

For every curve y = x^n, the gradient for every point on these curves is (n-1)x^(n-1).

The entire process is called "differentiation", and the end result is called "the first order differential".

The math peeps didn't stop there.

What if you apply the process again?

I mean what happens if you "differentiate" the "differential"?

For example: if y = x^5, the "first order" differential is 5x^4, by the rule above.

Let's call this D1

If we differentiate 5x^4 we get 20x^3, which is called the "second order" differential.

Let's call this D2.

But what happens if we know neither D1 nor D2, nor even y?

We have a "second order differential equation", which looks like;

A*D2 + B*D1 + C*y = 0

(A, B and C are just numbers)

The whole process of doing the reverse hinges on finding a version of y that, when differentiated twice, is proportional to y itself. In principle , it is quite simple.

That may all be very weLL, but what does it all mean?

Most of math is about abstract concepts, and this seems like no exception, but in fact, this concept has applications throughout the whole of science.

More about that next time.

NOW we start!

This blog is rapidly approaching its 4th birthday, and as no-one reads it, there is no reason why I should not get a little more involved with the science.

Here is the first post, as opposed to the last post.

What is a differential equation? Those few who have read before might have had a taste of the Calculus, which describes mathematically the rate at which things increase (or decrease) using stuff called "differentials". (look it up in the index under "maths", May 28, 2007).

Here is the simplest:



There are a number of terms here that need to be identified:

f(t) = a "function"; a mathematical way of explaining something (example: 2t + 1 is always going to equal 3, if t is one, or five if t is two, etc. This makes "2t + 1" a "function")

So what is "y"?

"y" is the end result, and the be-all and end-all of this thing is to get a "y = ......"

What is "dy/dt"? It is pronounced "dee-why by dee-tea", and is the rate at which "y" changes as "t" does.

Since we do not know what "y" is yet, these equations are far more difficult to solve than "2t + 1 = 7" (what is "t" here?).

The answer to these kind of equations is not a number, but another function

The simplest answer to the above differential equation is this:



...which is a mathematical statement saying that "y" is actually an exponential function (its value increases by multiplying the previous one by the rate it is increasing).

The "A" is just any old number, and the squiggly bit before f(t) indicates a process by which you can get another "function" from the original f(t).

Most differential equations are of this type, when solved (in other words, you get a "y = ....") give "e" to the power of something (e is 2.718281828...).

The reason "e" is involved is because the definition of "e" is that the rate of change of "e" in e**t with respect to t is "e" itself.

Comments welcome! ;)

Next is "Partial differential equations"...

Tinikling the Avarice of Silence...

...is a pun on the new blog title.

"Tinikling" is the National Dance of the Philippines. It involves ankles with poles.

"Silence" is because I have had only three comments in all of last month. I assume nobody reads this, but since I like writing anyway....

250 bonus points for each meaning you can see in this new blog title.

There are five, excluding the obvious one.

Can you get them all?





This is me, waiting for suggestions....

The blog title has now changed, so here are the answers.

"Tickling the Ivories of Science"

meanings:

1. "Ivories" refers to piano keys, often featured here.

2. "Ivories" also refers to "Ivory Towers" of mathematics.

3. "Tickling" refers to the way I approach the subject matter, as there are many allusions to "tickle"

4. "Science" refers to my own subject.

5. "Ivories of Science" means the best bits bits of science

6. What do you think Ben was doing when the photo was taken?

Agnetha, Benny, Bjorn, Anni-Frid

There was a time when these guys' humanity was put into question, simply because they seemed to be able to do no wrong.

Hit after hit after hit.

They had relationships inside the band, but this was their last greatest hit, in 1981.

"The Winner Takes it All" is a thinly veiled bittersweet account of their last days, when their relationships went sour, especially for Agnetha, the singer here.

Great song, but poignant simply because it can be interpreted on two different levels.

Agnetha, Benny, Bjorn, Anni-Frid...

A B B A

The boys went on to write the music for the musical "C H E S S", in 1986. It ran in London until 1989 at the Prince Edward Theater, near Cambridge Circus.

More Buzzword Bingo

These phrases on the left are apparently serious efforts of office people trying to communicate something meaningful, plus their actual meanings. A recent Local Government initiative in the UK banned these words. Sense at last has prevailed!


Across-the-piece - everyone working together

Actioned - do

Advocate - support

Area based - in an area

Area focused - concentrating on the area

Autonomous - independent

Baseline - starting point

Benchmarking - measuring

Best Practice - best way

Blue sky thinking - thinking up ideas

Bottom-Up - listening to people

Can do culture - get the job done

Capacity - ability

Capacity building - enough room in the system

Cascading - why use at all?

Cautiously welcome - devil in the detail

Challenge - problem

Citizen empowerment - people power

Client - person

Cohesiveness - together

Collaboration - working together

Commissioning - buy

Community engagement - getting people involved

Consensual - everyone agree

Contextual - background

Core developments - main things that are happening

Core Message - main point

Core principles - beliefs

Coterminosity - all singing from the same hymn sheet

Cross-cutting - everyone working together

Cross-fertilisation - spreading idea

Democratic legitimacy - voted in

Democratic mandate - elected to put people first

Dialogue - talk/discuss

Direction of travel - way forward

Distorts spending priorities - ignores people's needs

Early Win - success

Embedded - set in

Empowerment - people power

Enabler - helps

Engagement - working with people

Engaging users -- getting people involved

Enhance - improve

Evidence Base - research shows

Exemplar - example

External challenge - outside pressures

Facilitate - help

Fast-Track - speed up

Flexibilities and Freedoms - more power to do the right thing

Framework - guide

Fulcrum - pivot

Functionality - use

Funding Streams - money

Going forward - in the future

Good Practice - best way

Governance - Why use at all?

Guidelines - guide

Holistic - taken in the round

Improvement levers - using the tools to get the job done

Incentivising - incentive

Income Streams - money/cash

Indicators - measurements

Initiative - idea

Inspectorates - monitoring bodies

Interdepartmental - working together

Interface - talking to each other

Joined up - working together

Joint working - working together

Level playing field - everyone equal

Leverage - influence

Localities - places/town/city/village

Lowlights - worst bits

Meaningful consultation- talking to people

Meaningful dialogue - talking to people

Mechanisms - methods

Menu of Options - choices

Multi-agency -- many groups

Multidisciplinary - many

Municipalities - towns/cities/areas

Normalising - make normal

Outcomes - results

Outcomes - focused

Output - results

Outsourced - privatised

Parameter - limits

Participatory - joining in

Partnership working - working together

Partnerships - working together

Place shaping - creating places where people can thrive

Pooled budgets - money

Pooled resources - time and money

Populace - people

Potentialities - chances

Practitioners - experts

Preventative services - protecting the most vulnerable

Prioritization - most important

Priority - most important

Process driven - shouldn't everything be people driven?

Procure - buy

Procurement - buying

Promulgate - spread

Proportionality - in proportion

Protocol - guidance

Quick Hit - success

Quick Win - success

Rationalisation - cut

Reconfigured - reform

Resource allocation - money going to the right place

Revenue Streams - money

Risk based - safest way

Robust - tough

Scaled-back - cut/reduce

Scoping - work out

Seedbed - idea

Service users - people

Shared priority -- all working together

Signpost - point in the direction of

Single conversations - talking to

Single Point of Contact - everything under one roof

Situational - situation

Slippage - delay

Social contracts - deal

Social exclusion - poverty

Stakeholder - other organisations

Step Change - improve

Strategic - planned

Strategic priorities - planned

Streamlined - efficient

Sub-regional - work between councils

Sustainable - long term

Sustainable communities - environmentally friendly

Symposium - meeting

Tested for Soundness -- what works

Thematic - theme

Third sector - charities and voluntary organisations

Toolkit - guidance

Top-Down - ignores people

Trajectory - route

Tranche - slice

Transformational - change

Transparency - clear

Upward trend - getting better

Utilise - use

Value-added - extra

Vision -- ideal/dream/belief

Visionary - ideal/dream/belief

Welcome - necessary and needed/step in the right direction

Wellbeing - healthy

Worklessness - unemployed

SOFT- (loud)

This is the one that almost does not belong here.

Yundi Li.

I did not want to put this here since it is so seemingly NOT science. However, I have known this piece since I was about four years old. It is one almost every pianist knows by heart (without reading it, that is. Like a tune you whistle to automatically)

Regretfully, though, it IS science, simply because Mr Yundi does not even have to look at his hands. His eyes seem to know more, even when he is not using them.

The director is brilliant because he films the performer's hands in the reflection of the manufacturer's logo "STEINWAY AND SONS", the greatest piano constructors ever.

My professors make an analogy with physics people - The material is so familiar that no effort is required.

I know almost every note of this - it is played all across the World, and Yundi plays every note on time and in all the right places.

(It seems I am always chasing rainbows, but this takes the biscuit)

I have just noticed - he makes an error on the pedal, just before the trio.

Yundi is absolutely GORGEOUS. ( Sorry Ben )

Spirit of Love

This movie I have loved ever since I first saw it

Alan Tam is gorgeous and he does the singing also.

It is tragic, funny, ghostly, and a doomed love story.

I have seen it many times but I do not have a copy of it.

"Esprit D'Amour"

"Yam Yeung Cho"

Joyce Ngai, Alan Tam.

It is basically a tragic love affair with a beautiful ghost.

Alan Tam is the confused insurance clerk (Koo-Chi Min).

Joyce Ngai, the girl (See Yiu).

The theme is phenomenal. The song is called "Illusion".

At the beginning, he nearly runs down a beautiful girl in the rain.

Just after, she is tragically killed in a bizarre skateboard accident.

The insurance company decides it was suicide. It was not.

Her spirit returns, to obtain justice for her daughter, with a beautiful theme.

The clerk and the spirit fall in love.

She returns to claim the insurance money for her child.

The company refuses to pay.

The Spirit screws up the company in amusing ways.

BUT his fiancee finds out that he is in love with a spirit.
and takes steps, involving a "Door Of Death" through which the spirit must pass through.

After the spirit is exorcised, he tries to follow her by going through the "Door of Death", thus trying to commit suicide, by throwing himself out of the same window , saying "See Yiu - wait for me" (in Cantonese)

He sadly survives

At the end , after he recovers, he looks up at the sky. In the rain, glad to be alive, but glad to have known her Spirit. And then a girl in a car nearly runs him down, just like he nearly ran someone else down on a similar, rainy night...

CAN SOMEONE AT LEAST GET ME THE DOWNLOAD OF THIS?????

Saddle Up!

Way back in the years of when, it was speculated that order was needed in math, and geometry was the first victim.

A guy named Euclid decided he would formalise it, so he set out to "axiomise" it....

...and he came up with these, seemingly "self-evident" truths:

Euclid's Five Postulates are:

1. The shortest distance between any two points is a straight line

2. Any straight line can be extended (produced) indefinitely

3. Any point can be surrounded by a circle of any size

4. All right angles are equal to all other right angles

5. Parallel lines never meet

Above you see a translation of them into normal language. The first four don't need much tinkering with, but the fifth was originally stated thus:

"It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles."

Quite a mouthful, and clearly, in that context, NOT "self-evident".

Was there a way around this seeming contradiction? Apparently there still is not.

Quite a while after "Euclid's Fifth", several different people came forward with different interpretations of it. It turned out there were three. Einstein's general Relativity is based on one of the two others.

The three interpretations are really easy to state, but all of which are mutually exclusive:

Either,

a. parallel lines will always be parallel (Euclid),

or,

b. parallel lines will always meet (Riemann et al),

or,

c. parallel lines will always diverge (Riemann et al)

It was by using these "alternate" geometries, that Einstein came up with General Relativity in 1915.

From this acorn of an idea, there sprouted formalised geometry and a little thing explaining in full the anomaly of the precession of the perihelion of the planet Mercury to the tune of 43 seconds of arc per century.

Loopy

This is called a Klein bottle. It has no inside or outside. I use the concept to insult certain dumb individuals that infest the internet chat rooms with their smug arrogance.

Here's the party game you can play.

Announce a race - a cutting race. The participants have to cut a long loop of newspaper into two separate long loops of newspaper.

You need three people...

....and three long loops of newspaper.

Hana! Dul! Se! Go!

And the victims frantically start cutting, trying to be the first to produce two separate loops of newspaper.

We have a winner!

You raise his arm as the victor, then inspect his loops. Oh dear, there must be some mistake - you only have ONE giant loop! Where is the other one? Disqualified!

The second to finish is proclaimed the winner and her hand is held aloft.

We have a winner!

Let's just check your handiwork....hang on, you are supposed to get two SEPARATE loops - yours are linked like a chain! Disqualified!

The last to finish is now approached, and the disgruntled announcer inspects his work, and YES! he has two separate loops.

A WINNER!

Of course, the real winner is the math. Ten bonus points if you figure out how it's done.

137 million if you can construct a Klein bottle

Genius

WARNING SUBJECTIVE POST!!

This man is an absolute genius.

The album (Tubular Bells II) was recorded with just him playing ALL instruments, rerecording it time and again, in his own studio.

Take a look at how many musicians (keyboards, etc) are needed to perform the same thing LIVE.

John Gordon Sinclair joins Viv Stanshall (RIP) and Alan Rickman (Hans from Die Hard)in the announcement of the instruments. The post person spelled "Edinburgh" wrong

Note the change in guitar at the end. Wonderful! Note also the way he plays the guitar. Side swiping the strings, not up and down (Space Invaders???) and unique.

That GUT feeling

For centuries, well, decades, at least, mankind had striven to find a theory of "everything". That is, one single math formula that describes all of physics.

That goal is still some way off, despite massive strides that have been made towards it.

The argument goes something like this:

A total of four different forces have been identified. Please note as opposed to the previous post, these are specific kinds of force, whereas the previous post dealt with the CONCEPT of a force.

These four "types" of force hold the Universe together. However, they are somewhat limited in various ways.

In detail, these are they:


The Oldest known, Gravity. It acts on everything and is always attractive.

The next to be studied was Electricity, or as it is now known, Electromagnetism. This acts only on stuff that is electrically charged. That is, stuff that is not neutral electrostatically. We do not know what charge "is" yet, even today, just that stuff with the same "charge" repel, and stuff with opposite "charge", attract. There are only two of them

The third force is called the "strong" force, is always attractive and acts only on stuff that is desperately close.

The fourth is the "weak" force (we ran out of names long ago) and only acts on radioactive stuff (it causes atomic nuclei to split up)

There are no others

Each of these "forces" has it's own set of equations that describe only it (electromagnetism has four of its own)

It has long been a dream of all science to unite all these equations into one.

It has not been done yet.

They all seem to have two things in common, though. They all have a specific "range" and all use "exchange particles" to be felt. That is except gravity. Gravity has infinite range, but no "exchange particle" has ever been detected.

There may be hope, though. In 1974, Steven Weinberg and Abdus Salaam (I forget the name of the third), managed to marry the "weak" force with electromagnetism, calling it the "electroweak" force. The prediction, if this were true, would be three new "exchange" particles called the W+, W- and the Z-0. It wasn't until 1983 that we had the particle accelerator powerful enough to produce them, as they were predicted to be very heavy (E = mc squared, after all). Sure enough, they turned up.

So we had three forces now, not four.

There have been reliable equations suggesting that the "electroweak" can be married to the "strong" force, although to prove this we would need an accelerator the size of the galaxy to detect these exchange particles (dubbed the "X" particles)

However, there was another prediction from this marriage - proton decay. Protons are to all intents and purposes, stable, although the other one, the neutron, decays in about 16 minutes.

If the Electroweak and the Strong forces, were really part of a bigger picture, the proton would have a half-life of about 100000000000000000000000000000000 years (1 followed by 32 zeroes).

Since the universe is only about 15000000000 years old, none would have gotten round to decaying yet, UNLESS we watched 100000000000000000000000000000000 protons for a year.
Us peeps have been doing this for maybe a decade now, and so far no results (it is very hard to detect just one amongst all those, even without red herrings coming from just about everything else.)

If we do detect proton decay, we will only have gravity to deal with.

That is the be-all and end-all. Gravity, almost like an unusually stubborn mule, blatantly refuses to conform.

Ok, so there may be a way out by conjecturing "m-theory", "brane-theory", and "string-theory", but to prove this requires travel in eleven dimensions. In other words, these are only words, not proof.

Warning! SUBJECTIVE STUFF!

Gravity differs because it is geometric as refined by Einstein. The other three are gauge FIELDS, and require exchange particles to be felt.

Geometry by definition does not require exchange particles and I am surprised that even Einstein tried to meld his own theory with the other established forces. Maybe it was peer pressure.

END OF SUBJECTIVE STUFF

Suffice it to say, we do not really have a Grand Unified Theory (GUT) yet, although we may have one for everything except Gravity.

The whole picture can be summed up in an energy diagram (see below), as we believe all the forces were one once, when there was so much energy there was no distinction. As the Universe expanded from the Big Bang, the amount of energy decreased (per cubic meter) and the forces separated in a process known as "symmetry breaking". Th.e diagram shows that gravity is conjectured to be there, but that is more like hope than speculation

We try to address this by putting lots of energy into our particle accelerators to maybe recreate conditions as they were during the big bang.



We still have a GUT feeling one is out there somewhere. Lord knows, we've tried!