Gini and Stolarsky means in
geometric problems
Alfred Witkowski
University of Technology and Life
Sciences, Bydgoszcz, Poland
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
What is n-frustum?
Truncated cone with n-dimensional object as its
base:
Trapezoid is an 1-frustum
El Castillo in Chichen Itza is a 2-frustum
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Problem
How does the n-volume of selected horizontal
sections (s) depend on n-volumes of its bases (x,y).
Case n=1 was considered by Howard Eves in
Means Appearing in Geometric Figures, Math.
Magazine, 76, 4, (2001), 292-294
x
s
y
𝑥𝑟 + 𝑦𝑟
𝐺 𝑟, 𝑠; 𝑥, 𝑦 = 𝑠
𝑥 + 𝑦𝑠
1 (𝑟−𝑠)
𝑠 𝑥𝑟 − 𝑦𝑟
𝐸 𝑟, 𝑠; 𝑥, 𝑦 =
∙
𝑟 𝑥𝑠 − 𝑦𝑠
1 (𝑟−𝑠)
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Cylinder with the same
(n+1)-volume and height
x
s
y
s
Formula discovered (in case
n=2) in 50 BC by Heron of
Alexandria, that’s why we call
them Heronian means.
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Frusta of equal (n+1)-volumes
x
x
s
s
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
s
y
Equal heights
x
x
s
s
`
s
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
y
x
Similar frusta
x
s
s
s
y
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Equal lateral volume
x
x
s
s
s
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
y
Centroid
(center of mass of solid frustum)
x
s
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Center of mass of bases
(or „inner” cones of equal (n+1)-volume)
x
x
s
s
y
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Similar „inner” cones
(or intersection of „diagonals”)
x
x
s
y
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Lagrangean point
Point where gravitational attraction
of x cancels that of y
x
s
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
(n+1)- volume of frustum equals
sum of (n+1)-volumes of cylinders
x
s
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Cylinders of equal lateral volume
x
s
y
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Order of means (n>3)
n>3
Lagrangean point
Similar inner cones
Similar frusta
Equal heights
Cylinder of the same volume
Vol two cylinders=vol frustum
Centroid
Frusta of equal lateral vol.
Frusta of equal volume
Equal lateral vol of cylinders
Centers of masses
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Order of means (n=3)
n=3
Lagrangean point
Similar inner cones
Similar frusta
Equal heights
Cylinder of the same volume
Vol two cylinders=vol frustum
Frusta of equal lateral vol.
Centroid
Frusta of equal volume
Equal lateral vol of cylinders
Centers of masses
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Order of means (n=2)
n=2
Similar inner cones
Lagrangean point
Similar frusta
Equal heights
Cylinder of the same volume
Vol two cylinders=vol frustum
Frusta of equal lateral vol.
Centroid
Frusta of equal volume
Equal lateral vol of cylinders
Centers of masses
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Order of means (n=1)
n=1
Similar inner cones
Similar frusta
Lagrangean point
Equal heights
Cylinder of the same volume
Frusta of equal lateral vol.
Vol two cylinders=vol frustum
Equal lateral vol of cylinders
Centroid
Frusta of equal volume
Centers of masses
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary
Homework
CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary