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So, I have this geometry:

enter image description here

It was created from this geometry:

enter image description here

by simple scaling to zero about the 3D cursor, which had been moved to selection, but selecting the topmost and bottom-most vertices. Now, as you may already know, this isn't nearly as connected as it looks; the fact all these edges are intersecting doesn't mean there is a vertex there to unite them. Let's try to fix that with:
enter image description here

But no dice, it becomes impossible to do loop cuts. This:

enter image description here
becomes impossible after the merge. Could someone be so kind as to tell me why? Is there a way around this problem, or should I just do my loop cuts first?

Doing the loop cuts first seems to work, but I'd like to understand the nature of the problem.

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The fundamental problem is that loop cuts work on quads; they are interrupted by triangles, and faces with more than four edges. You can do the loop cut before you merge all the points together, because Blender treats each face as a quad, that is a face defined by four vertices, where one edge has zero length because two of the vertices have the same co-ordinages. The merge makes all of quads triangles, because two of the vertices on each face are combined into one. Similarly, if all of the three or more vertices of a face have the same co-ordinates, you still have a face, even though the face has zero area.

With respect to achieving an effect similar to the loop cuts, switch to edge select mode, select all the radial edges (but not the circumferential ones), and use the subdivide tool, setting the number of subdivisions to the desired value. Another way is to use the knife tool. Yet another way is to extrude the vertices around the circumference, and scale them almost to the center, then extrude again, and scale the second set of vertices to the center point, and merge the center vertices together. Now you can still make your loop cuts in the quad faces created by the first extrusion, and you can select the edge loop created by the second extrusion and use the edge slide to position it where you want it.

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  • $\begingroup$ So the fact that one edge has length 0 does not mean blender sees it as a point.. $\endgroup$ – user1833028 Jun 22 '15 at 23:27
  • $\begingroup$ @user1833028 Blender doesn't care about if virticies are in the same place. We need to take care of them using the remove doubles operator if we want to get rid of them (usually you don't want doubled verticies) $\endgroup$ – TARDIS Maker Jun 23 '15 at 16:44
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You can add edgeloops on a circle with a triangles fan in 2 simple steps:

First select all the faces of the circle and inset it using I button, position the edgeloop in the middle of the circle radius and hit Enter.

Then select the edgeloop and bevel it using Ctrl+B shortcut, then by dragging the mouse you can adjust the distance between the edgeloops and with the mouse wheel you can adjust the amount of edgeloops.

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    $\begingroup$ Nice. Another quick way to do it is to use circle-select while in Edge Select Mode and you will instantly have all the "spokes" selected. Then you can simply subdivide into as many parts as you like. Alternatively you could, in Vertex Select Mode, select the center vert and then grow the selection using [Ctrl +] (and subdivide). $\endgroup$ – Mentalist Jun 23 '15 at 3:09

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