SUO: The Plumbing Theorem(s)
To answer the questions about dyads and triads as simply as
possible, I would like to explain the issues in terms that
anyone can verify just by going down to the basement and
examining the plumbing.
Let us suppose that you had the following resources:
1. Long lengths of water pipe and the option of ordering
more whenever you need it.
2. The ability to cut the pipe to any desired length and to
finish the ends with whatever threading is needed to link
it with suitable connectors.
3. A large supply of dyadic connectors: sleeves, which let
you link two pipes in a straight line; and elbows, which
let you link two pipes at an angle (usually a right angle,
but other angles may be permitted).
With those resources, you can direct water from the point of
entry into your house to at most one faucet elsewhere in the
house. You cannot direct the water to any additional faucets,
bathtubs, showers, or toilets.
But if you were given some triadic connectors (T shape or
Y shape), you could connect one additional faucet or other
facility for each triadic connector you use.
Connectors with more than 3 links would be unnecessary.
A tetrad (+ shape) could be used instead of two triads.
But if you had enough triads, dyads, and straight pipe,
you wouldn't need any tetrads, pentads, hexads, etc.
All these facts can be verified by looking at the plumbing,
playing with pipes, or drawing diagrams until you are satisfied
that they are true.
John Sowa