It's not just Cubes Marching
With the cubes running, a block would expand to those around it. In fact, with the default settings for the Marching Cubes algorithm, the result is a Rhombicuboctahedron, shown below.
That is the result of considering the eight vertices of a set of cubic blocks as input for the gear cube algorithm.
It's Cubes Marching
We do not need to establish cubes and see which vertices are covered or exposed. Instead, we can establish vertices directly.
See the video gear cube algorithm for a demonstration. Note that the set of vertices used as input for the gear cube algorithm does not form cubes. Instead, there are only three vertices (one isolated, two contiguous).
The linked video shows how the following entry:
Results in the following output:
As you can see, we end up with an octahedron (similar to the one shown in the question) that comes from the isolated vertex. In addition to an elongated square bipyramid that comes from the two contiguous vertices.
See the 15 settings of the original Marching Cubes article:
The tetrahedron is formed by the second configuration of the 15 shown above (the one with a single highlighted vertex), which is applied to the eight cubes surrounding the isolated vertex.
The other algorithms
I want to draw attention to the fact that Marching Cubes is not the only solution. There are also Octahedra marching and Tetrahedra marching. Unfortunately, information about these is less abundant.