Math CentralQuandaries & Queries


Question from gabrielle, a student:

In general how many planes are there which contain two given points, three given points, and four given points?

We have two responses for you

Hi Gabrielle.

Two distinct points define a line.

Three points that are not all on the same line (in other words, not colinear) define a plane. This is why a table with three legs (reasonably constructed) doesn't fall over and why a tripod supports a camera. The plane is the floor in either case.

If you have four points and they are not on the same plane (not coplanar), then they define a space. So the simplest Platonic solid is a tetrahedron (a four-sided die), which has four corners.

So how many planes contain two given points? An infinite number. How many contain three given (non-colinear) points? One. How many planes contain four (non-coplanar) points? Zero.

Stephen La Rocque. >


There is a system to this - and an analogy with lines in the plane.

First the analogy (a lot of mathematics is first grasped using analogies).

In the plane, two points make a line.
So for two points there will 'in general' be one line.
For three points 'in general' there will not be a line.
For one point (stepping down) there are an infinite number of lines, one for each 'direction' creating what could be called a fan of lines (technically called a plane pencil of lines).

Now for 3-space and planes.

Three points 'in general' (not collinear, chosen at random, ... ) will form a triangle, and this will all fit a unique plane.
Four points (like the corners of a tetrahedron or a triangular pyramid) will not all be on any plane, though triples of them will form four different planes.
Stepping down, two points form a line, and there wil be a fan of planes with this line (like pages of an open book, with the line down the spine of the book).

Walter Whiteley
York University

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