**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

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 **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:**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*...

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