Calc Avatar: The Last Airbender Feats - Book Two: Earth Collection Part 3

^The Feat


Appa = 157 pixels
Wan Shi Tong Library Spiral Section = 75 pixels
Wan Shi Tong Library Spire Top Diameter = 177 pixels

Wan Shi Tong Library Spiral Section/Appa = 0.478
Wan Shi Tong Library Spire Top Diameter/Appa = 1.127

Appa = 10.256 meters (here)
Wan Shi Tong Library Spiral Section = 4.902 meters
Wan Shi Tong Library Spire Top Diameter = 11.559 meters


Wan Shi Tong Library Spire Top Diameter (Red) = 15 pixels
Wan Shi Tong Library Foundation 1 Radius (Green) = 306 pixels
Wan Shi Tong Library Foundation 1 Height (Blue) = 52 pixels
Wan Shi Tong Library Foundation 2 Radius (Purple) = 293 pixels
Wan Shi Tong Library Foundation 2 Height (Orange) = 36 pixels
Wan Shi Tong Library Building Radius (Bright Orange) = 215 pixels
Wan Shi Tong Library Building Height (Teal) = 499 pixels

Wan Shi Tong Library Foundation 1 Radius/Wan Shi Tong Library Spire Top Diameter = 20.4
Wan Shi Tong Library Foundation 1 Height/Wan Shi Tong Library Spire Top Diameter = 3.467
Wan Shi Tong Library Foundation 2 Radius/Wan Shi Tong Library Spire Top Diameter = 19.533
Wan Shi Tong Library Foundation 2 Height/Wan Shi Tong Library Spire Top Diameter = 2.4
Wan Shi Tong Library Building Radius/Wan Shi Tong Library Spire Top Diameter = 14.333
Wan Shi Tong Library Building Height/Wan Shi Tong Library Spire Top Diameter = 33.267

Wan Shi Tong Library Foundation 1 Radius (Green) = 235.804 meters
Wan Shi Tong Library Foundation 1 Diameter = 471.608 meters
Wan Shi Tong Library Foundation 1 Height (Blue) = 40.075 meters

Here and here from Legend of Korra appear to corroborate some kind of square foundation at any rate. The angles suck to scale from, but it gives a vague idea of the depth the schematic doesn't *shrugs*

Volume = L^2*H

L = Wan Shi Tong Library Foundation 1 Diameter
H = Wan Shi Tong Library Foundation 1 Height

Wan Shi Tong Library Foundation 1 Volume = 8,913,245.284 m^3

Wan Shi Tong Library Foundation 2 Radius (Purple) = 225.782 meters
Wan Shi Tong Library Foundation 2 Diameter = 451.564 meters
Wan Shi Tong Library Foundation 2 Height (Orange) = 27.742 meters

Volume = L^2*H

L = Wan Shi Tong Library Foundation 2 Diameter
H = Wan Shi Tong Library Foundation 2 Height

Wan Shi Tong Library Foundation 2 Volume = 5,656,872.499 m^3

Wan Shi Tong Library Building Radius (Bright Orange) = 165.675 meters
Wan Shi Tong Library Building Diameter = 329.824 meters
Wan Shi Tong Library Building Height (Teal) = 384.533 meters

Instead of measuring each part of the building individually, I'm just going to treat the building part of the Library as a 80% Hollow Half Ellipsoid.

Volume = 0.2((2/3)PIr1^2r2)

r1 = Wan Shi Tong Library Building Diameter/2
r2 = Wan Shi Tong Library Building Height

Wan Shi Tong Library Building Volume = 4,418,918.171 m^3

Total Wan Shi Tong Library Volume = 18,989,035.954 m^3
Rock Density = 2,700 kg/m^3
Wan Shi Tong Library Mass = 51,270,397,075.8 kilograms


Obviously, Toph is out of scale with the spiral portion of the Library's spire. For sake of internal consistency, Toph related to the spire is irrelevant.

Wan Shi Tong Library Spiral Section (Red) = 830 pixels
Start (Green) = 24 pixels
End (Blue) = 124 pixels
Distance = Start - End = 100 pixels

Distance/Wan Shi Tong Library Spiral Section = 0.12

Distance = 0.588 meters
Time Frame = 0.042 seconds (24 FPS, 1 Frame between, here and here)
Library Fall Speed = 14 m/s

KE = 0.5mv^2

m = Wan Shi Tong Library Mass
v = Speed

Toph Beifong's Earthbending Yield = 5,024,498,913,429 joules or 1.201 kilotons

Mind you, this is just treating the feat as if its Free Fall. Wan Shi Tong was forcing the Library down through an unknown layer of supporting sand or even bedrock to place it entirely beneath the dunes.

Such compressive forces would necessitate pressures so great the sand would basically be turned into glass the same way carbon is compressed into diamond.

Sand basically being Silicon Dioxide? That's a pretty substantial Latent Heat of Fusion.

For example? If the sand the Library was sitting on was only even about 1 meter thick? Across a surface acea covering the Library's 451.564 meters of Foundation? Sand density being 2,334 kg/m^3? You'd be looking at about 475,926,047.364 kilograms of Silicon Dioxide. Being compressed into Glass? Through just the Latent Heat of Fusion (159,800 j/kg) alone ignoring temperature change entirely?

Gets you 18.177 kilotons for both Wan Shi Tong and Toph Beifong's Earthbending. You can definitely argue upwards of the top tower's height above the sand for layer of sand/rock the Library needed to be pressed through to sink. This model is highly speculative though and doesn't really tell us how the Library ended up in the Spirit World, upside down, hanging from a fucking bunch of tree vines.