“Enrichment beyond the classroom”
GREECE and ITALY
join a Mediterranean math-collaboration!
etwinning project 2015-2016
2.
Mission: Find out the hidDen Min-Max!
WHERe is it ? Part 2-3
THE soap bubble activity
2.
First watch the video we created about the ... "soap bubbles and the hidden math"!
Then you can find out more details what is about and study about the Min-Max problem known as Steiner-tree problem.
A brief explanation about the video activity:
Soap bubbles and steiner's reticles
Visualization of math subject!
How intresting!
"Let us find help by soap bubbles through this experiment with the large ring and lanyard , it will form a circular hole in the centre because outer foil to minimize surface tension , will determine a maximum hole area with same perimeter ."
And now some ...more Math!
Move to the next step of the prezi presentation using arrows.
"Make a soap bubble and observe it :
you could spend a lifetime studying it "
( Lord Kelvin )
Mission: MATH and SOAP BUBBLES Schimatari
A soap bubble was originally a mixture of olive oil and ashes, born in Syria 3000 years ago ,then Marseille became its largest manufacturing center in the fifteenth century. It is actually a thin layer of soapy water that forms a sphere with an iridescent surface.
A bubble can exist because a liquid has a certain surface tension , that is, it does involve similarly to an elastic sheet . In any case , a bubble made from by pure water is not stable and it is necessary to dissolve the soap to stabilize it .
In water , in fact , the surface tension is too strong , the bubble is unstable and breaks down immediately.
The soap and all its derivatives have as a common effect to decrease the surface tension of water , consequently , when the surface of the film bubble is going stretching and thinning, because a bubble is inflated , the concentration of soap in the film decreases and this causes an increase of the surface tension . So the soap goes to reinforce the weakest parts of the bubble and tends not to enlarge it further. The soap also reduces the evaporation allowing the bubble to last longer , although this effect is not forever .
Whatever the distortion of its original shape, a soap bubble will always be spherical.
This happens because the bubble , under the action of surface tension , will assume the lowest possible surface according to the physical principle of minimization.
For a given volume of air , for example the one we blown , the shape of the container with the smallest surface is precisely the sphere .
The biosphere designed by the architect Renzo Piano in Genoa, Italy
When you find that the foundation of a glorious and ancient cities of North Africa depends on an ox and the mathematical knowledge of a Phoenician princess, you can not help but be surprised . Carthage, the bubbles , the igloos ... no relationship between these things ? Yes , a strong bond and geometrical important optimization problems ...
According to the story of the Aeneid of Virgil ( Roman poet of the first century BC ) ,
the legend of the foundation of Carthage (now a suburb of Tunis) said that Dido ,
the Phoenician princess forced to flee the wrath of his brother Pygmalion who had killed
her husband , fled to the African coast ( during the ninth century B. C. ) where he bought
from the local king of Numidia, a land in order to establish with his people.
The king, who diplomatically refused every new settlement in their country , and it was
believed clever , only agreed to grant the plot of land that could be covered by a single ox
hide !
Dido exploited the most of it : he cut the ox hide into thin strips and arranged in rows ,
forming a long rope ( measured between 1000 and 2000 meters ), which he then fitted on
the ground so as to enclose the largest area possible, (ask students to make assumptions,
let us find help by soap bubbles) in the form of a circle.
It may be noticed that if the circle has a surface area Maximum for an equal perimeter ,
spilling reasoning is obtained that circle has a perimeter Minimum for an equal area.
We visualized it by Geogebra:
To tell the story of the isoperimetric problem one must begin by quoting Virgil:
At last they landed, where from far your eyes
May view the turrets of new Carthage rise;
There bought a space of ground, which Byrsa call’d,
From the bull’s hide they first inclos’d, and wall’d.
(Aeneid, Dryden’s translation)
Let us find help by soap bubbles through this experiment with the large ring and lanyard , it will form a circular hole in the centre because outer foil to minimize surface tension , will determine a maximum hole area with same perimeter .
A translation attempt
Greek to English
Ίσως η παλιότερη ιστορία εξαπάτησης όπου χρησιμοποιήθηκαν μαθηματικά την αναφέρει ο Βιργίλιος στην «Αινειάδα» του. Η ιστορία της Διδούς . Η Διδώ ήταν πριγκίπισσα της Τύρου, που πήγε στη Βόρεια Αφρική και ίδρυσε την Καρχηδόνα.
Η Διδώ είχε κληρονομήσει τον θρόνο της Τύρου από τον πατέρα της, Ο νεότερος όμως αδελφός της Διδούς, ο Πυγμαλίωνας, δολοφόνησε τον συζηγο της Σιχαίο και κατέλαβε την εξουσία. Μόλις το έμαθε η Διδώ, παρέλαβε τους θησαυρούς του νεκρού πλέον συζύγου της και επιβιβάσθηκε σε ένα πλοίο μαζί με μερικούς αφοσιωμένους της Τυρίους και δούλους της. Το πλοίο τους μετέφερε στην Κύπρο και από εκεί στις ακτές της Λιβύης, στη χώρα Γετουλία ή Νουμιδία, όπου ζήτησε από τους ντόπιους και τον βασιλιά τους Ιάρβα να της επιτρέψουν να χτίσει στην ακτή μία πόλη. Ο Ιάρβας αρχικώς αρνήθηκε, όταν όμως η Διδώ του προσέφερε πλούσια δώρα δέχθηκε, υπό τον όρο η πόλη να καταλαμβάνει τόση έκταση όση ένα τομάρι βοδιού.
Η Διδώ τότε έκοψε το τομάρι σε πολύ στενές λωρίδες και, ενώνοντας τη μία με την άλλη, περικύκλωσε τόσο χώρο, ώστε της έφθασε να κτίσει την Καρχηδόνα και την ακρόπολή της, τη Βύρσα (βύρσος = δέρμα, τομάρι).
Η Διδώ εφάρμοσε αυτό που όλοι όσοι ασχολούνται με την γεωμετρία γνωρίζουν και είναι το αρχαιότερο ίσως πρόβλημα μεγιστοποίησης:
Από όλες τις καμπύλες του επιπέδου που έχουν το ίδιο μήκος, αυτή που περικλείει χωρίο με το μέγιστο δυνατό εμβαδόν είναι ο κύκλος.
Γνωστό ως
ισοπεριμετρικό πρόβλημα.
Perhaps the oldest deception story where used mathematics says Virgil in the 'Aeneid' of. The story of Dido. Dido was princess of Tyre, he went to North Africa and founded Carthage.
Dido had inherited the throne of Tyre by her father, the younger brother of Dido, however, the Pygmalion, murdered spouse of Sichaio and seized power. Once learned, the Dido, received the treasures of the now deceased wife and boarded a ship with some devotees of cheese and slaves. The ship was transferred to Cyprus and from there to the coast of Libya in Getoulia country or Numidia, where requested by the locals and their king Iarva to enable it to build on the shore a City. The Iarvas initially refused, but when Dido offered him rich gifts received, provided the city to occupy as much area as much as an ox hide.
Dido then cut the skin in very narrow strips and joining one another, surrounded as much space, so the arrived to build Carthage and its citadel, the Vyrsa (vyrsos = skin, pelt).
Dido applied what everyone involved knows the geometry and is perhaps the oldest maximization problem:
Of all the level curves of the same length, that includes passage of the maximum possible area is a circle.
known as
isoperimetric problem.
because...
for each student
"Math Investigation" eTwinning project
was designed by the two mathematician teachers
to be the "place" where
through cirriculum activities
he can let
his imagination drive towards his creativity
using his own skills and intrests.
GEOGEBRA
activities
Watch the video we formed to be familliar to GEOGEBRA...
Try to reconstruct the steps...
3.
click on the video to watch...
