Typically space filling blocks are rectangular/cubic. On a
horizontal surface the blocks stack nicely because: a) the
top surfaces of the blocks are also horizontal, and b) gravity
is perpendicular to this.

I am interested in non-cubic space filling blocks.
Non-cubic
space-filling blocks do not have the benefit of gravity
sticking the blocks together. The blocks must be connected
by another mechanism.

I was first interested in rhombic dodecahedrons. My interest
has now moved to golden rhombohedrons. Golden
rhombohedrons are interesting because they can be stuck
together to form rhombic dodecahedron and rhombic
triacontahedrons. They are also interesting because they
can be stacked in 3 dimensions in a non-periodic
configuration, thus making them a good model of quasi-
crystals.

I have discovered there was a building toy with the shapes
of golden rhombohedron called "Rhombo". Each face of each
block contained a magnet, allowing the blocks to be
connected and thus forming interesting structures. As far as
I can tell it is no longer for sale.

Using magnets as the connector for these blocks is an
obvious choice since it allows the faces of the blocks to be
smooth so the blocks can be slide together. This is especially
important when putting in the last piece in a concave space
(e.g. the last piece in a rhombic triacontahedron).

Anyway, I thought it would be interesting to come up with a
surface connector that had similar functionality as magnets.

So my idea is a surface that contains a rotating connector.
The rotating connector is attached to a spring that, when
activated, causes the connector to move from a flat
orientation to an angled orientation (see illustrations).

Two connectors when placed against each other, will rotate
around approximately the same axis, and connect.

The connectors are initially held in a flat configuration by a
resilient finger (not shown in illustrations). The resilient
finger is just strong enough to resist the pull of a tension
spring. When pressure is applied to the surface of the
connector (by another connector placed on top), this
disengages the resilient finger, and the tension spring pulls
the connector into the angled configuration.

One surface will require 2 (or more) of these connectors
which will form a 'dovetail' configuration. With enough force
the connector can be pulled apart. The connectors can then
be reset by pushing the connectors into the resilient fingers
in the flat configuration.

How did I not know about golden rhombohedrons?!
As for your connectors, I've pondered the same dilemma
(but pertaining specifically to connecting identical robots
together, although I'm after rotational symmetry).
Perhaps if the connector had a mass at the "non-spring" end,
then if you "whack" them together, the masses will activate
the connector when the rest stops moving on impact.
If you set up your hinge & spring so it's an "over-centre"
alignment (only just past in the "flat" position, obviously),
you can do away with the "fingers" of your design.

It's just saying if a golden rhombus (diamond shape in more
colloquial terms) has a width (from the two opposite corners)
of 1cm then the length (from the two other opposite corners)
will be
1.618cm. If it has a width of 10cm then the length will be
16.18cm, etc.

The golden ratio, phi, (for some reason normally given as
"1.618..." (ie. bigger than 1) instead of the usual "ratio"
value of less than 1...) is found in various places in
nature, as well as (apparently) "aesthetically pleasing".
But that's just artsy waffle...
It has the unique property that 1/phi = phi - 1, AND
(actually mathematically equivalent) phi^2 = phi + 1.
ie: 1/1.6180339887 = 0.6180339887
and 1.6180339887^2 = 2.6180339887
It's also the limit of the ratio of 2 consecutive values of
the Fibonacci sequence.
(Yes, I'm a maths geek. Among other things, I did an
essay/study of Fibonacci for a 3rd-year university maths
paper...)

PS: "phi" is the Greek letter phi, normally written in
Greek...

PPS: phi = (1+sqrt(5))/2
(the easy way on a calculator is 1.25, sqrt, +, .5)

Another way to imagine it is this: imagine a line divided into two
unequal parts, where the ratio of the whole to the larger part is
the same as the ratio of the larger part to the smaller part.

//the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller part.//

Aha! I can see it now. I frustrated the hell out of my teachers in school because I could only learn visually at that time so although math was my worst subject and I failed miserably I totally aced things like trigonometry and geometry because I could see them in my head. Bring my mark back up to a nice 52% or so overall so I could attend the next grade.

Except circle math. I never really understood exactly what Pi was until I watched a three second gif a few years ago. It was just another number to be memorized by rote.

Seriously... time-machine, past guidance-counselors, ass-kickings all-round.

//I was first interested in rhombic dodecahedrons. My
interest has now moved to golden rhombohedrons. Golden
rhombohedrons are interesting because they can be stuck
together to form rhombic dodecahedron and rhombic
triacontahedrons. They are also interesting because they
can be stacked in 3 dimensions in a non-periodic
configuration, thus making them a good model of quasi-
crystals.//

I was struck by 2 things in this paragraph. How
"Deliciously nerdy" it sounds AND "That sound's interesting
to me".

Squeeze together connection system using lower air pressure? but if the shapes are too interlocking, it might not be too easy to get the matching faces apart.