Spira mirabilis

L'espiral logarítmica, també coneguda com espiral equiangular, de creixement o spira mirabilis, és una corba molt present a la natura i que té la particularitat de no canviar de forma en augmentar la mida. Es defineix, en coordenades polars, per la següent equació

en què r és el radi, a i b són escalars i θ és el valor de l'angle entre el radi i la tangent en un punt determinat.

(A la imatge es poden veure els punts P i P2, els radis que hi passen en negre i les tangents en rosa.)

Com es pot observar, els dos angles definits en cada cas són iguals, complint una de les propietats d'aquestes espirals: l'angle θ és constant. De fet, una definició d'espiral logarítmica correspon a la "corba que talla tots els radis a angle constant". Un cas especial és la circumferència, que és una espiral de radi 90 graus.

L'espiral logarítmica és present a molts fenòmens naturals: les seves propietats permeten un empaquetament òptim, de manera que la relació entre mida i espai ocupat sigui la més eficient possible. D'això es poden veure alguns exemples a la natura: el creixement dels embrions humans, dels mol·luscs, de les pinyes del pi, de les llavors a les flors com el gira-sol...

Va ser Descartes qui va descriure l'espiral logarítmica primer, però a Jakob Bernouilli se li acrediten extenses investigacions i el nom que encapçala aquest escrit, spira mirabilis ("espiral miraculosa" o "meravellosa").

Referències:
Spira Mirabilis
Logarithmic Spiral - Wikipedia
Where can we find equiangular spirals?

Riddle

‘Twas whispered in Heaven,
‘Twas muttered in Hell,
And echo caught faintly
The sound as it fell;
On the confines of Earth,
‘Twas permitted to rest,
And the depth of the ocean
Its presence confessed;
‘Twill be found in the sphere
When ’tis riven asunder,
Be seen in the lightning
And heard in the thunder.
‘Twas allotted to man
With his earliest breath,
Attends him at birth
And awaits him at death,
Presides o’er his happiness,
Honour and health,
Is the prop of his house
And the end of his wealth.
In the heaps of the miser,
‘Tis hoarded with care,
But is sure to be lost
By the prodigal heir;
It begins every hope,
Every wish it must bound,
It prays with the hermit,
With monarchs is crowned;
Without it the soldier,
The sailor may roam,
But woe to the wretch
Who expels it from home!
In the whisper of conscience
‘Tis sure to be found,
Nor e’en in the whirlwind
Of passion is drowned;
‘Twill soften the heart,
But though deaf to the ear,
It will make it acutely
And instantly hear.
But in short, let it rest
Like a delicate flower,
Oh, breathe on it softly,
It dies in an hour!

Catherine Maria Fanshawe

Roses are red...

Roses or Rhodonea curves... well worth the while and the effort!

A Song of Love

Say, what is the spell, when her fledgelings are cheeping,
That lures the bird home to her nest?
Or wakes the tired mother, whose infant is weeping,
To cuddle and croon it to rest?
What the magic that charms the glad babe in her arms,
Till it cooes with the voice of the dove?
'Tis a secret, and so let us whisper it low
And the name of the secret is Love!
For I think it is Love,
For I feel it is Love,
For I'm sure it is nothing but Love!

Say, whence the voice that, when anger is burning,
Bids the whirl of the tempest to cease?
That stirs the vexed soul with an aching - a yearning
For the brotherly hand-grip of peace?
Whence the music that fills all our being - that thrills
Around us, beneath, and above?
'Tis a secret: none knows how it comes, how it goes -
But the name of the secret is Love!
For I think it is Love,
For I feel it is Love,
For I'm sure it is nothing but Love!

Say, whose is the skill that paints valley and hill,
Like a picture so fair to the sight?
That flecks the green meadow with sunshine and shadow,
Till the little lambs leap with delight?
'Tis a secret untold to hearts cruel and cold,
Though 'tis sung, by the angels above,
In notes that ring clear for the ears that can hear -
And the name of the secret is Love!
For I think it is Love,
For I feel it is Love,
For I'm sure it is nothing but Love!

Lewis Carroll

Problema III: el berenar

Sabem que un got de llimonada, 3 galetes i 7 pastissets valen 14 pessetes; i que un got de llimonada, 4 galetes i 10 pastissets valen 17 pessetes:

1) Quant valen un got de llimonada, una galeta i un pastisset?

2) Quant valen 2 gots de llimonada, 3 galetes i 5 pastissets?

Va, que aquest és interessant. Teniu uns dies abans no doni la resposta! Noteu, però, que a la pregunta 1) no es demana el valor separat de cada cosa sinó els tres a l'hora. Així doncs, aquí teniu, perquè aneu agafant ganeta (i esteu agraïts que no us demani quant val el que hi ha a la fotografia!):

One day typical

Yesterday it was Monday and the endof was finished. I was made dust after two days of not giving stick in the water, and the bed was so warm, that the sheets stuck to me and I stayed myself in the bed. My father hit me a yell and it went of little that I don't fall myself from my bunk. I was at dot of arriving late to the school. Less damage that the teacher didn't say "You have an unpunctuality"!

When I went out to the yard, it was making a terrible wind, and I said to my friend "Look what wind it is making!" and my friend responded "You have reason!". Half an hour more late, the bell touched and a repellent boy said to me "Give yourself hurry, you go late!" and I said "Go to fry asparagus, piece of donkey!" and, followedly, I went to the class.

I passed three hours whole in class, and at the halfday I sat myself at the side of the wall. I made some duties and studied one little, and I ate the eaten one.

For the afternoon we made an exam, which went to me well, and then came a teacher of guard. I passed myself super well, and it missed little that one boy fell for the window!

It arrived the hour when the school finished, and I went to pick the bus. I was black because the two boys behind me were striping me. One hour later I got down of the bus and I went home at darks. I made the rest of my duties, ate my supper and put myself in the bed, at the elevens and half.