A semi-cube is ½ of [0, 1]n divided thus:
The plane Σxj = n/2 bisects the cube [0, 1]n.
Q = {x | 1≤j≤n → xj>0}.
H = {x | Σxj < n/2}.
B = Q∩H.
B is a simplex and its centroid is the centroid of its n+1 vertices.
If z is a vector and S is a set then S + z = {s+z | s ∈ S}.
For n=3 I want the centroid of the semi-cube on the origin side of the plane.
That semi-cube is
Q ∩ ∼((Q+(0,0,1))∪(Q+(0,1,0))∪(Q+(1,0,0))) ∩ H.
Moment-1 about the origin of the semi-cube is its centroid times its volume which is ½.
B is a simplex and its centroid is the centroid of its vertices = ¼(3/2, 3/2, 3/2) = (⅜, ⅜, ⅜).
(A)
B’s volume is (3/2)3/6 = 9/16.
(A)
Moment-1 of B is (27/128, 27/128, 27/128).
(Q+(0,0,1)) ∩ H is a simplex, similar to B, with vertices at (0, 0, 1), (0, 0, 3/2), (0, ½, 1), (½, 0, 1).
Its centroid is at (⅛, ⅛, 9/8).
The centroids of the three congruent simplexes are (⅛, ⅛, 9/8), (⅛, 9/8, ⅛), (9/8, ⅛, ⅛).
Their mutual centroid is (11/24, 11/24, 11/24).
Each volume is 1/48 and their total volume is 1/16.
Their collective moment-1 is (11/384, 11/384, 11/384).
The moment-1 of the semi-cube is the difference = (35/192, 35/192, 35/192).
The semi-cube’s volume is ½ and thus
The semi-cube’s centroid is (35/96, 35/96, 35/96).
The floating cube’s centroid is (½, ½, ½) and is at the water line.
root(j)*(sl - (m OF s)[1]/c OF s)
summit | WL bottom
0 c sl 1
--------------
vol centroid moment-1
B 0.5625 .375 0.2109375
3b 0.0625 0.4583333333333333 0.028645833333333332
SC 0.5 0.3645833333333333 0.18229166666666666
(sqrt (* 3 (s (- 35/96 1/2)))) => 0.23454854685828547 (distance of SC centroid above water)