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Cone Symmetry


Date: 7/17/96 at 4:34:34
From: Kang hui min
Subject: Symmetry for a cone

Hi! Why is it that a circular cone has infinite lines of symmetry?

Thanks!


Date: 9/27/96 at 16:47:12
From: Doctor Jerry
Subject: Re: Symmetry for a Cone

Dear Kang hui min,

I'll try to answer your question by first asking what does it mean for 
an object O to be symmetric relative to a plane P?  This is like a 
mirror image of an object.  For each point of the object in front of 
the mirror, there is a corresponding point in the mirror image.  The 
object and the corresponding point are at equal distances from the 
mirror but on opposite sides of the mirror. So, for an object O to be 
symmetric relative to a plane P, for any line L perpendicular to P, if 
(a,b,c) is a point belonging to both L and O, then there will be a 
second point of O on L and at the same distance from P as the first 
point.

I'll assume that an object O is symmetric relative to a line LL if for 
any line L intersecting LL and perpendicular to LL, if (a,b,c) is a 
point belonging to both L and O, then there will be a second point of 
O on L and at the same distance from LL as the first point.

With this definition, a right circular cone is symmetric relative to 
the plane through its vertex and perpendicular to its axis. It is also 
symmetric relative to any plane containing its axis.  A right circular 
cone is symmetric to its axis.  It is also symmetric relative to any 
line passing through the vertex and perpendicular to the axis of the 
cone.  There are infinitely many such lines.

-Doctor Jerry,  The Math Forum
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Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Symmetry/Tessellations

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