The Pentagon is the best shape for a large buildings as long as all surrounding buildings are also Pentagons
Who needs straight roads anyway. We would have 3 way intersections instead plus the wind would have less of a impact
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Progressing through the regular polyhedra from triangles to squares to pentagons to hexagons, pentagons are the only ones that don’t tile a plane. How do you figure they’re the best for buildings?
pentagon
plane
oh no
e: well I expected this low-effort comment to get buried within seconds so just to be clear, it's purely a mathematical theory where if you had two pentagons and an extra vertex, then only 9/11 of the edges would tesselate correctly. I'm not sure what that is in Radians, but it's definitely an American 77 units for sure.
I expected this low-effort comment to get buried within seconds
Designated park space
But the Earth isn't a plane.
Sure, human scaled patches of the Earth's surface can be approximated by a similarly sized patch of a plane, but if we're talking about tiling the entire surface of the Earth with buildings, it can actually be done using twelve pentagons or twenty isosceles triangles. We just need buildings whose footprints are roughly 1/12th and 1/20th the Earth's surface respectively.
For the pentagon, that'd be around 510.07 × 1012 m² divided by 12 = 42.505 × 1012 m². With the Pentagon building having seven floors, one such building would have roughly 297.541 × 10^12 m² of floor space.
For the triangle, that'd be around 510.07 × 1012 m² divided by 20 = 25.503 × 1012 m². Assuming this building has seven floors like the Pentagon building does, it'd have roughly 178.524 × 10^12 m² of floor space.
The good thing about dividing into triangles, however is that it can be subdivided into four similar isosceles triangles, which can be applied recursively down to a far more realistic scale.
Doing that, we can subdivide the original triangles sixteen times yielding the following:
25.503 × 1012 m² / (416) = 5.937 × 10^3 m²
And since the area of an isosceles triangle is equal to s²(√3)/4 we can rearrange things to find the side length of a compound with area of 5 937m²
s = √(4A/(√3)) = 117.103 m
I think that's a human-enough scale for buildings.
In total, there's 85 899 345 920 such buildings, covering the Earth.
If one such building has 7 floors, it'd have at most 41 559 m² of floor space.
EDIT:
Hit enter too soon. Additional proofreading.
Damn, I discovered a small mistake in the calculations partway through. Corrected.
That makes me think of the last episode of Severance, where two characters who have lost their memory of the external world are debating whether the equator is a continent or a continent-sized building.
My high ass was like ‘wait they put tile on aircraft?’ and after accepting that I just assumed it messed with airflow or the Bernoulli effect.
Then I was like, ‘Oh’.
Technically it's only regular pentagons that don't tile the plane.
I know everything about planes because I am shorter than my brothers. When we were young, they called me Tattoo and if I pointed at something with my finger, they screamed in unison, "Da plane! Da plane!"
They line up in zig zag patterns
Oh
My shower thought was way off