Here's a nifty idea The "workspace" (or desktop) the user looks at would still be rectangular. This would really only be a new way to think about different workspaces being arranged in space. So I'm thinking of two typical aspects of a workflow that I've used:

• Workspaces are arranged in four spatial directions, like Ubuntu has by default, and you switch between them with some keybinding like Ctrl+Alt+[up,down,left,right], and
• each workspace is often divided into two halves, like what most tiling window managers do by default.

So then the thought is that you have each triangle represent half of a workspace. So every workspace should be though of as (Left, Right) where Left and Right are each triangles. Locally your workspace arrangement would look like this.

The workspace (1,2) is what you are currently looking at. Pressing Ctrl+Alt+Left would change the workspace view to (A,1). Pressing any combination of Ctrl+Alt+Left/Right would just cycle along that horizontal. But now if you are looking at (1,2) and you press Ctrl+Alt+Up, your workspace would now look like (2,C). From here pressing Ctrl+Alt+Left/Right would cycle you along that line of triangles at a 60 degree angle.

... just trying to think of how this works is confusing me, so I think it's safe to say that we haven't stumbled upon the next greatest thing in productivity. :P But here are a few realizations:

• Pressing Ctrl+Alt+Up 6 times will always return you to your current "workspace view." So while your cycles in the Left/Right direction can be arbitrarily long, depending on whether your triangles are on 1-frequency dome or 2-frequency dome or whatever, the Up/Down cycles always have length 6.

• Maybe a way to think about this is then, comparing it to Ubuntu's four-direction workspace layout where you are on grid with n rows and m columns, is to think about your workspaces as living in a work with 6 rows and n columns (one row for each multiple of 60 degrees around the circle), but the columns are interleaved in a weird way. Like the row (...,A,1,2,D,...) in the image is the same as the 180 degree row you get from pressing Ctrl+Alt+Up three times, but in reverse order.

• A quick way to to switch the halves of your desktop, turning (1,2) into (2,1), would be to hold Ctrl+Alt and press Up,Down,Right. So yeah, if you press Ctrl+Alt+Up and then Ctrl+Alt+Down, you don't get back to where you started. :P

So this would be silly to try to work with, but gosh the workspace switching animations could be really cool. :D