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Wednesday, 28 July 2010

Axons get a kiss

Axons get a kiss

Wow, just wow.  A quick one today, I wanted to share this photograph of neurones firing into action after treatment with a chemical; this sheds a whole new light on the spectacular action of our nervous system and the firework display that occurs up close,  each and every time one of our nerves fire into action.  This is phenomenal, I think you'll agree!

"Axons get a kiss" Photograph by Dr Christine Jasoni, Department of Anatomy
and Structural Biology, University of Otago.
Basal forebrain axons (red) and GnRH neurons (green) following treatment
with Kisspeptin.

Saturday, 17 July 2010

Who the heck is The Vitruvian Man?

This famous drawing in ink on paper is a diagram of geometrical ideology, a showcase of exceptional penmanship and a microcosm of the workings of the universe.  Sound too good to be true? Well you had better believe it; the amalgamation of artistic and scientific objectives is exemplified by the Vitruvian man, which was created way back in 1487 during the renaissance.  Wow, Da Vinci, your foresight astounds. 

The Vitruvian Man, drawn by Leonardo Da Vinci, depicts two men superimposed upon one another with his arms and legs apart, inscribed in a circle and a square. The drawing is based on the ideal human proportions with geometry described by architect and first-known engineer, Vitruvius, in De Architectura.  Leonardo’s interpretation strays slightly from Vitruvius’ specifications as he combined his reading of the text with his own examination of actual human bodies.
'The Proportions of the Members in a Human Body'

So here is a guide to the ‘Ideal’ male Proportions: A palm is the width of four fingers; a foot is the width of four palms, a cubit (forearm length) is the width of six palms and a pace (length of a full stride measured from when one heel lifts from the ground and when the same heel hits the ground again at the end of the step) is four cubits.

I know you have already been measuring so here’s a couple more that are a touch more difficult to measure – A man’s height is four cubits (forearms!) or 24 palms. The length of his outspread arms is equal to his height.

(For more average proportions and an examination of the precise geometry,
this is a fantastic article )

The major difference between Da Vinci’s drawing and those that had gone before him was the addition of the circle and square inscribing the two superimposed positions. The centre of the circle cannot be the same as the centre of the square which demonstrates the way in which man’s centre of mass (or centre of magnitude, the central point in the body) changes dependent on position yet his centre of gravity (imagine a line of symmetry dissecting him down the middle) stays the same.
This artwork is used around the world to symbolise mans closeness to nature, man as the measure of all things and has numerous other interpretations. The symbol is used by medical and holopathic wellness companies, on one side of the Italian euro coin and on a NASA patch given to astronauts that have completed a spacewalk.

The Vitruvian man exemplifies nature’s perceived perfection, intriguing patterns and impeccable symmetry. Pretty cool, huh?

Tuesday, 6 July 2010

What's the deal with Spirographs?

It’s the heady days of the 1990s.  You’re 8 years old, hair teased into plaits and it’s raining outside.  You’ve grown bored of buckaroo and no one will play scrabble with you so you’re mother (who has a vested interest in expanding your young, neuroplastic brain) whips out the Spirograph. 
You poke your Disney pencil in one of many small plastic holes in the small gear and push it round and round inside the larger gear creating patterns that, at that age, were just darn ‘pretty’; but at the ripe age of 21, we see they are much more than that, they are hypotrochoid mathematical curves.

There is a surprisingly complex mathematical basis for the graphs that are produced and without wanting to give you a stroke, take a quick peek at this handy equation that explains the maths behind these ‘pretty’ shapes.

x = (R+r)*cos(t) - O*cos(((R+r)/r)*t)
y = (R+r)*sin(t) - O*sin(((R+r)/r)*t)
(moving circle outside the fixed circle)

x = (R-r)*cos(t) + O*cos(((R-r)/r)*t)
y = (R-r)*sin(t) - O*sin(((R-r)/r)*t)
(moving circle inside the fixed circle)

Yeah, I know, and I’m sorry, but I don’t expect you to be able to relate to that jumble of letters.  Basically, Imagine a fixed point set inside one circle that travels along the inside of a second circle. The complex harmony of the position of the point and the diameters of the two circles create a complex looping track.  The different hypotrochoids are determined by the differing diameters of the two circles and the position of the point inside the centre circle where the line is drawn from. 

This might help - You can use this applet to change the dimensions of each of the circles to get a better understanding of how the maths affects the resulting shape!

Thanks to Anu Garg for permission to use this applet

arc generated by a circle traveling along the inside of another circle is not only ruddy beautiful, it is also ridiculously useful!  Guilloché patterns are ‘spirograph-like curves that frame a curve within an inner and outer envelope curve’. So, they are like spirographs but layered over one another.  They are used on banknotes, securities, and passports worldwide for added security against counterfeiting.  If you look closely at any British bank note you will see this pretty pattern in the background.

Phew, I think that’s enough for today.  If you are more Newton than Monet, you can have an epiphanic mathematics-induced high at this page of hypotrochoid equations to wrap your head around.