Awhile back, when Dick was talking about pouring concrete,
I was having discussions with Glenn Stockton, a local, who
grew up working on a dam, the one near Page, Arizona.
That must have been Glen Canyon dam -- easy mnemonic.

This morphed into an imaginary / fantasy setting, could be
a videogame, where we had Earthians and Martians working
together on such a dam, each with a camp on alternate sides
of the chasm. A rope goes across and "buckets" of concrete
get sent out over the dam, then lowered, payloads dropped.

That's all an excuse to focus on "units" of concrete, and
these large floating beach ball looking things that waft back
and forth across the chasm. These giant balloons represent
a "shared concept" between the Earthians and the Martians.

The Earthians take the spherical balloon's radius as a unit
for the edge of their cube. The RxRxR cube is what they call
a unit of volume. This is how they measure their concrete.

The Martians take the spherical balloon's diameter as a
unit for the edge of their tetrahedron. The DxDxD tetrahedron
is what they call a unit of volume. This is how they measure
their concrete.

What I dwelt on in earlier installments of this story was the
size relationship between the RxRxR cube and the DxDxD
tetrahedron: the former is 1.06066+, relative to a latter volume
of 1, or in more Earthian terms, the cube is clearly One, or
One Unit, so the Martian tetrahedron of edges 2 has the
smaller volume of 0.94281-.

What I've been learning from David Koski lately is how to
look at that Martian tetrahedron of edges 2, as if it were two
flappy triangular "book covers" with a shared spine. The
"wing tips" of these covers are what get closer together
(to the point of touching) or further apart (to the point of
being at 180 degrees from one another w/r to the "hinge"
or "spine"). The Richard Hawkins Youtube at grunch.net
is a good place to start, if you're having trouble visualizing.
The "flapping tetrahedron" in the first few seconds in what
I'm talking about.

If the regular tetrahedron of all edges 2 is if smaller volume
than the RxRxR cube, then might we open the covers a bit
wider, increasing the distance between tips, to get a volume
that's precisely the same? Indeed we can, and it turns out
the angle in question is 90 degrees. The two book covers,
each one a 2x2x2 triangle, place at 90 degrees to each
other, with a line between their tips, has a volume equivalent
to that of the RxRxR cube, or R3.

R3 = 1.6066 D3 = 1 Iceberg = 1 Book(90) = 1.6066

Book(90) is our 2-triangles "book" or "butterfly" open at
90 degrees, with a resulting wing span or 6th edge. It
has the same volume as a cube of edges 1/2 the length
(R) of its winged 5 edges (D). Only the 6th is a variable
in this toon.

David's vision goes further than this, into space-filling with
threesomes, and P, Q, and R modules -- a kind of genesis
story with the "book covers" tetrahedron morphing a bit,
such that the spine my also vary. I hope he will tell us
more.

In the meantime, I'm happy to have another point of contact
between the Earthians and Martians. The Earthians take
an all-edges-2 regular DxDxD tetrahedron from the Martians
and snip a wire, allowing two flaps to open up more and
become 90 degrees to each other. That's a volumetric
synonym for the RxRxR cube. For the Martians,
Book(arccos(1/3)) = 1 where arccos(1/3) is the dihedral
angle of a regular tetrahedron with all edges D.

Kirby