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.




2. How It Works: The Sewing Machine


It's a topological question: when the needle passes the tissue below, it carries the wire in the loop. On the other side there is a system that puts the wire of the second spool (the little one in the machine) in this loop, then the needle rises, taking with it the second wire.
In this way a series of knots are formed along the seam between the spool higher, the big one, and the lower, the small one.
The first practical and widely used sewing machine was invented by Barthélemy Thimonnier, a French tailor, in 1829.


1. Nights without stars

Imagine how it would be our idea of the Universe if the sky was always covered by clouds.
At a certain hour of the morning we would begin to have a diffuse light and after a while the dark returns. We would not have a reference that will show us the time during day or night (Sun, Moon or stars) and would not know even the shadows of the various objects exposed to the sun.
In addition to not know the charm of some sunsets, sundials would not have been invented.
We would have the same clues that could provide some element: magnetic materials that orients according to a certain direction, the pendulums that tend to have a rotary motion and the relation between the periods of the year where the days are warmer, with the fact that they are also brighter and longer.
Thanks to different position of stars at different latitudes, Greek philosophers were able to understand curvature of the Earth.
Or, as a result, the observation of the shadow of the Earth projected onto the Moon during a lunar eclipse.
And, finally, on the horizon that seems bent, with ships that disappear at a certain distance from the shore, providing a clear indication of the curvature of the Earth, perhaps we couldn’t observe it so frequently, since few people would venture offshore without the help of the stars for orientation.

The French physicist Gaspard Gustave de Coriolis in 1835 showed that an observer placed on a rotating frame of reference will have to deal with forces (Coriolis) which depend not only by angular velocity, but also by direction and body speed.
These fictitious forces are the basis of cyclonic or anti-cyclonic atmosphere systems formation.

To remember the sense of rotation of the phenomenon you can use this simple mnemonic scheme (valid only in the northern hemisphere):

·          Anticyclone (high pressure)           >>>     Clockwise

·          Cyclone   (Low pressure)               >>>     Counterclockwise


The effects of this force are also manifested in atomic physics (in polyatomic molecules whose molecular motion can be described as a rigid rotation with in addition a vibration of the parts around the position of equilibrium) and this produces a particular confusion, in the molecular spectrum, between the rotational and vibration levels.
Flies (Diptera) and some moths (Lepidoptera) exploit the Coriolis effect in flight with specialized appendages and organs that relay information about the angular velocity of their bodies. Coriolis forces resulting from linear motion of these appendages are detected within the rotating frame of reference of the insects' bodies. In the case of flies, their specialized appendages are dumbbell shaped organs located just behind their wings called halteres. The halteres oscillate in a plane at the same beat frequency as the main wings so that any body rotation results in lateral deviation of the haltered from their plane of motion. In moths, their antennae are responsible for the sensing of Coriolis forces in the similar manner as with the halters in flies. In both flies and moths, a collection of mechanic-sensors at the base of the appendage are sensitive to deviations at the beat frequency, correlating to rotation in the pitch and roll planes, and at twice the beat frequency, correlating to rotation in the yaw plane.

The best known example is the Foucault pendulum.


It is a pendulum free to swing in any direction for many hours. The Foucault pendulum was first presented to the public in 1851 and consisted of a sphere of 28 kg suspended on the dome of the Pantheon in Paris with a 67 m long wire. In an inertial system, it would track lines always in the same direction, but it didn’t.
At different latitudes of the Earth, except along the equator, it’s observed that the oscillation plane of the pendulum slowly rotates. In particular at North and South Pole the rotation is in a sidereal day: the swing plane remains stationary while the Earth rotates, in accordance with the law of motion of Newton.
At other latitudes the oscillation plane rotates with an inversely period proportional to the sine of the same latitude; at 45° the rotation occurs every 1.4 days, at 30° every 2 days and so on.

The Coriolis force is also used in modern gyroscopes that use solid state switches with MEMS (Micro-Electro-Mechanical Systems).

What is certain is that without the help of the stars the world-view of the Universe would be definitely different.



What you will find in this blog is the result of reflections on scientific topics known and less-known, of links and references ... so a hotchpotch of ideas.
Many descriptions and sources quoted in the posts are taken from Wikipedia, The Free Encyclopedia; in any case, the authors of quotes and links can ask me to be removed them from the blog or, if you do need them, to quote freely this site.
The various posts are only intended to present the most various disparate topics that will be explored via the links in the text.
In summary a free flow of ideas.