9. Avatar Flower

Dini’s surface has the interesting feature of having constant negative curvature that can be created by twisting a pseudosphere:


It is named after Ulisse Dini and described by the following parametric equations:



The flowers that appear in the movie "Avatar" seem to have the same shape:



8. Hyperspheres

As reported in wikipedia:  http://en.wikipedia.org/wiki/Hyperspherea hypersphere is the set of points at a constant distance from a given point called its center.

The "volume" of a hypersphere is given by:

where Γ denotes the gamma function.

The "surface area" is instead given by:

In three dimensions, the formula for the calculation of volume is:

As for the surface is:

For the definition of derivative, as the limit of the quotient, we have:
the surface is the derivative of the volume.

And this in each dimension:     D (Vn) = Sn-1

Their relationship in every dimension is remarkably simple:

    Vn / Sn-1 = R / n       e.g.  in  3  dimensions we have:  V3 / S2 = R / 3


For the calculation of hypervolumes and hypersurfaces, formulas contain:
in 2 or 3 dim. appears π,  while in 4 or 5 dim. π2,  6 or 7 dim. π3  and so on.

As an example for even number of dimensions, we have:


while for odd number of dimensions:



6. Wallis and Pippenger Infinite Products

The Wallis product is a formula found in 1655 by John Wallis, which calculates the simple infinite product:

Nick Pippenger obtained an infinite product for  e:


5. Tesseract

In geometry, the tesseract is the four-dimensional analog of the cube.

Hypercube is composed of:

8 Cells (or Cubes),  24 Faces,  32 Edges  and  16 Vertices

and it meets the extension of Euler's formula:   

Faces + Vertices = Cells + Edges       

There may be different types of representations; such as this where all edges have the same length:


Or a central projection:


In four dimensions, the hypercube is also called tesseract (from the greek τέσσερις ακτίνες or "four beams").
A famous example of a tesseract is the Arch of La Défense, a monument located in the modern district of La Défense in Paris. The official name in French is Grande Arche de la Fraternité.






4. Vincent – Starry Nights

And if the cloud bursts, thunder in your ear
You shout and no one seems to hear
And if the band you're in starts playing different tunes
I'll see you on the dark side of the moon.
                                                Roger Waters - Brain Damage



Starry, starry nights
Paint your palette blue and grey
Look out on a summer's day
With eyes that know the darkness in my soul
Shadows on the hills
Sketch the trees and the daffodils
Catch the breeze and the winter chills
In colors on the snowy linen land

Now I understand
What you tried to say to me
And how you suffered for your sanity
And how you tried to set them free
They would not listen, they did not know how,
Perhaps they'll listen now

Starry, starry night
Flaming flowers that brightly blaze
Swirling clouds in violet haze
Reflect in Vincent's eyes of china blue

Colors changing hue
Morning fields of amber grain
Weathered faces lined in pain
Are soothed beneath the artist's loving hand

Now I understand
What you tried to say to me
And how you suffered for your sanity
And how you tried to set them free
They would not listen, they did not know how,
Perhaps they'll listen now

For they could not love you
But still your love was true
And when no hope was left inside
On that starry, starry night
You took your life as lovers often do
But I could have told you, Vincent,
This world was never meant for one as beautiful as you

Starry, starry night
Portraits hung in empty halls
Frameless heads on nameless walls
With eyes that watch the world and can't forget
Like strangers that you've met
The ragged men in ragged clothes
The silver thorn, the bloody rose,
Lie crushed and broken on the virgin snow

Now, I think I know
What you tried to say to me
And how you suffered for your sanity
And how you tried to set them free
They would not listen, they're not listening still
Perhaps they never will.
                                           Don McLean - Vincent


Vincent van Gogh produced more than 900 paintings and 1,100 drawings, despite the fact that he lived only 37 years. He suffered from a form of epilepsy and a manic tendency to write and paint. In January 1889 he was discharged from the hospital in Arles, where he had been hospitalized after the incident. The precise chain of events that led to the celebrated incident of van Gogh slicing off his ear is not known reliably in detail.

In just over a decade he produced more than 2,100 artworks, including 860 oil paintings and more than 1,300 watercolors, drawings, sketches and prints.

But the artist felt that he had not fully recovered: his mental health was not yet integrated. So, in May, he sought asylum at the psychiatric hospital near Saint Remy in France.

It was here that he painted some of his most famous works, such as "Starry Night".


"Starry Night" is an oil painting on canvas (92x73 cm), made in 1889 and preserved in the Museum of Modern Art ( MoMA ) in New York.

The images shown below, indicate which constellations are supposedly depicted in the paintings and were taken from the online magazine Elapsus:


Starry Night

Starry Night Over the Rhone

Vincent (by Don McLean) is from the album American Pie recorded and released in 1971. It's also known as Starry Starry Night.

The words of the first stanza are a clear reference to the painting Starry Night.
It's possible to listen to the following YouTube link:




3. Rule of 9’s

The rule of 9% is used to estimate the extent of a burn.

Percentage of Body Surface in an Adult:

Head and Neck       9%
Upper torso         9%     (Also posterior 9%)
Mid  torso          9%     (Also posterior 9%)
Upper limbs         9%     (Each)
Lower limbs        18%     (Each)
Perineum            1%


Percentage of Body Surface in a Child:

Head and Neck      18%
Upper torso         9%     (Also posterior 9%)
Mid  torso          9%     (Also posterior 9%)
Upper limbs         9%     (Each)
Lower limbs        14%     (Each)


Rule of Palm:

Each palm is approximately 1% of Body Surface.