For a better idea of how the bond angles look, here is an energy-minimized structure of windowpane (4.4.4.4). Notice how it's (nearly) planar. That central carbon atom (middle gray dot, blues are hydrogens) is the one to watch.

Here is 5.5.5.6 windowpane:
![[5.5.5.6]fenestrane: InChI=1/C14H22/c1-2-10-4-6-12-8-9-13-7-5-11(3-1)14(10,12)13/h10-13H,1-9H2](/moleculeoftheday/images/5.5.5.6.fenestrane.gif)
And here is an unstrained compound, tetramethylmethane (2,2-dimethylpropane):

The changes are subtle but result in enormous energy changes. The molecules go in increasing order of stability (Decreasing energy). This is one of the things it's a little weird to start thinking about in chemistry. Molecules are rarely completely rigid; they're usually at least a little floppy, and the way we draw them isn't usually quite the right shape.
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How stable is that Windowpane and how would it fold up? Looking at your picture it looks like that central carbon would be an awesome oscilator back and forth, You could almost see the clock pulses or stimulated emission, but I have no idea if the corners would be the things that bend around and mess things up.
HyperChem minimizes tetrahedral neopentane to 2.2697 kcal/mole and puckered [4.4.4.4]fenstrane to 168.04 kcal/mole. All fenestrane carbons are assigned "cyclobutane" type.
http://ww.mazepath.com/uncleal/fenestr.png
Stereogram
The central carbon is not flat (four times 79.8669 degrees is 319.47) but it does have all four bonds on the same side.