70 years have passed

Meditations in the space time.
1945, 25th of April in Udine as well as in all the rest of Italy, the WW2, the second world war was officially finished bringing finally peace and freedom to the country.

In front of kids of the age of my parents, about 8 years old in that time, a parade of real soldiers, real weapons and people that suffered real wounds and lost many of their beloved was leading them out of the nightmare of war... entering then the illusion of peace in a democracy manipulated by international interests of the winners and cold-war!

2015, 25th of April, Udine. 
In front of kids of the age of my son, 7 years old now, a cosplay parade of puppets, fake soldier, warriors and monsters of all kinds is leading them out of their own reality into the dream illusion of freedom, world peace and justice following the main stream creativity of those who have been bombed the most... 70 years ago, the Japanese people. 

"Wannabes" feelings, machismo, idols, sexual winks of all kinds fill up the air with an unreal buzz dancing into the groove of this subtle but heavy media manipulation.
Fake wounds, plastic weapons, fancy colors wigs and fantasy costumes hiding the identity but revealing the need of being seen, becoming someone else at least for an afternoon, jumping on top of the stage in front of everybody...

Let our generation be the link of wisdom, 
reading the past with eyes unveiled by hatred, 
living in the present with strong fearless heart,
allowing the future to bloom into the human 're^Evolution of consciousness.
PEACE

Dome vs cube: the math of the results

In this post I would like to share the results of my mathematical meditations on the geometry of domes and cubes, considering the beauty of perfect conditions that all the proper rules give. It's all aimed at studying geodesic domes as an approximation of the half sphere.
Scratching my rusty memories a bit, especially about the properties of radicals (!!!), I managed to find out by myself all the results that I couldn't find elsewhere online. So I have been very proud of myself after so many years out of practice... I started examining the geometry of the dome as an half-sphere of radius R, in comparison with the half-cube of height a (thus with base 2a). So the basic formulas of volumes V and areas A are the following:

Then I solved the equations expressing the height of the half-cube a as a function of R, in the two cases of (1) the area of the half-cube being equal to the area of the half-sphere or (2) the volume of the half-cube being equal to the volume of the half-sphere. These two situations give completely different results but with some sort of "connection" and here they are:

the next step is to calculate in the first case (1) the volume of the half-cube and compare it with the volume of the half-sphere and in the second case (2) the area of the half-cube and compare it with the area of the half-sphere. These final steps have been by far the trickiest so that I had to keep checking the results over and over... before getting to something that resembled the expected outcome. I did both comparisons in the form of percentage proportion so in the first case of (1) the half-cube with the same area of the half-sphere, the volume of the half-cube is about 72.36% of the volume of the sphere. Here's the calculations.

In the second case of (2) the half-cube with the same volume of the half-sphere, the area of the half-cube is about 124.07% of the area of the half-sphere and here's the other calculations.

So at the very end the half-sphere (dome) wins on efficiency in both cases, encasing the biggest volume with a given area and the smallest area with a given volume! It all goes along with the rules of the Nature as they appear to be: we can all sleep well trusting that things are good, in the perfect world of theory...

Penta-hexagonal dome inspiration

In these latest nights of geodesic meditations, I have been exploring an apparently simplified dome model that is based on hexagons and pentagons, rather than triangles but basically just grouping them up.

Because of this, it turns out to be a little bit more sacred geometry alike, so it fits me way better. It's all just empirical reasoning and few hand drawn sketches but I feel this is a good concept.

The first model is actually made of pentagons and triangles, equivalent to the 2v frequency. The section in red is the fifth section of the ideal cover.

The next level up frequency that works with this concept is the 3v that gives a classic football-ball feeling. An interesting property of this design is that every joint connects always three corners, which is a fact that cannot be underestimated.

As the design grows in complexity it jumps up to 6v frequency, which is actually simplified down to only 4 patterns with 4 struts-length: one regular pentagon, one regular hexagon and two irregular hexagons. The lengths of the four struts (a,b,c,d) are calculated in a percentage of the longest strut (side of the regular hexagon). The number of each kind of pattern can be easily calculated: as we keep going up in complexity, the number of pentagons does not change (they're always six) as well as the number of regular hexagons (always ten, but just smaller and smaller compared to the radius of the dome). We just keep adding more and more kinds of irregular hexagons. The next levels up should be frequency 9v, 12v... always adding three extra: I didn't go further than this for now and probably it should be enough. 
The tricky point is calculating the correct angles of each irregular hexagon, planning to make each pattern flat. 
This is going to be the main subject of my next meditations.