CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Gini and Stolarsky means in geometric problems

Alfred WitkowskiUniversity 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-frustumEl Castillo in Chichen Itza is a 2-frustum

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Problem

x

y

s

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

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Cylinder with the same (n+1)-volume and height

Formula discovered (in case n=2) in 50 BC by Heron of Alexandria, that’s why we call them Heronian means.y

s

x

s

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Frusta of equal (n+1)-volumesx

y

s

y

s

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Equal heights

s

x

y

s

x

`

s

y

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Similar frustax

y

sx

s

y

s

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Equal lateral volumex

y

s

x

s

s

y

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Centroid(center of mass of solid frustum)

x

y

s

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Center of mass of bases (or „inner” cones of equal (n+1)-volume)

x

y

s

x

y

s

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Similar „inner” cones (or intersection of „diagonals”)

x

y

s

x

y

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Lagrangean pointPoint where gravitational attraction of x cancels that of y

x

y

s

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

(n+1)- volume of frustum equals sum of (n+1)-volumes of cylinders

x

y

s

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Cylinders of equal lateral volume

y

s

x

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Equal heightsCylinder of the same volume

Frusta of equal lateral vol.Vol two cylinders=vol frustumEqual lateral vol of cylinders

Equal heightsCylinder of the same volume

Frusta of equal lateral vol.Vol two cylinders=vol frustumEqual lateral vol of cylinders

n=1n=1n=2n=2n=3n=3

Order of means

Similar inner conesSimilar inner conesLagrangean pointLagrangean point

Cylinder of the same volumeCylinder of the same volume

Similar frustaSimilar frusta

Equal heightsEqual heights

Vol two cylinders=vol frustumVol two cylinders=vol frustum

CentroidCentroid

Frusta of equal lateral vol.Frusta of equal lateral vol.

Frusta of equal volumeFrusta of equal volume

Equal lateral vol of cylindersEqual lateral vol of cylinders

Centers of massesCenters of masses

n>3n>3

CIA'10, September 19--25, 2010; Hajdúszoboszló, Hungary

Homework