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I don't know. I'm not all that smart, but to me it seems that all they did was repeat dimensions 1-3 for dimensions 4-6, 7-9, and then returned for the 10th.
If you follow the illustrations, dimensions 1, 4, 7, and 10 are points. Dimensions 2, 5, and 8 are intersecting lines. Dimensions 3, 6, and 9 are folds. The 4th dimension is built from the 3rd, the 7th dimension is built from the 6th, and the 10th dimension is built from the 9th.
It seems too contrived to me.
I remember reading Cosmos some years ago. I prefer Carl Sagan's explanations of spatial dimensions much more than these explanations. Basically, to reach a dimension one higher than your current dimenstion, add a right angle to a point in the corner of the current dimension. So, to get from the 3rd dimension to the 4th, add a right angle to the corner of a cube.
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"I can normally tell how intelligent a man is by how stupid he thinks I am" - Cormac McCarthy, All The Pretty Horses
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