user2765: To find the degree (or the number of attacked squares), of a square's corresponding vertex for all pieces and their corresponding graphs, given any square of the board.  In other words, given a piece and a square - the number of attacked squares by the piece at the given square on an empty board is what the student finds.  This is repeated until all squares are examined.
Also, has anyone had the idea to make a 3-D representation for each piece, where each square has a respective height equal to the degree of its' corresponding vertex in the given piece's graph (where the degree is the number of squares attacked by each specific piece at each square)?  Of course, the rook's graph representation would be dull, as the graph is regular (ie the number of attacked squares is the same at any square),  but all the other pieces, save for the pawn, would give students the visual lesson that the pieces in the center have more options.  Not only this, it appeals to an old military adage - to control the high ground - which in these cases is the center, if at all possible.  

user11629: Not in 3D and not super-fancy, but it should get across the point. I did not do it for the rook because as you note the rook always has 14 squares to go to (on an empty board).
However if you want to use this to argue for putting pieces in the center, I'd be careful, because:

This is for an empty board. Lots of things can happen when other pieces are on the board.
The illustration for the king is probably of limited value because (unless you are Steinitz) we don't usually think of the king as an active piece.
For the queen and bishop (which are essentially the same, only shifted by 14, the number of rook moves) you notice that the difference between different squares (center and border) is not all that great. In my opinion a much more important factor for positioning queen and bishop centrally in a real game is that they have more options, i.e. a bishop in the center can go in four directions while a bishop in the corner has only one direction. This means that a bishop in the center can attack things in all four directions and likewise in order to block this bishop, the opponent has to block four directions. Also for transferring the bishop to other parts of a board a central position might be better.
For the knight there could be some value in this as it cannot be blocked by other pieces, so the situation for a knight is essentially the same as on an empty board. That's where you get the saying: "A Knight on the rim is dim" from. But as with all such rules there are plenty of exceptions.






2765: <p>To find the degree (or the number of attacked squares), of a square's corresponding vertex for all pieces and their corresponding graphs, given any square of the board.  In other words, given a piece and a square - the number of attacked squares by the piece at the given square on an empty board is what the student finds.  This is repeated until all squares are examined.</p>

<p>Also, has anyone had the idea to make a 3-D representation for each piece, where each square has a respective height equal to the degree of its' corresponding vertex in the given piece's graph (where the degree is the number of squares attacked by each specific piece at each square)?  Of course, the rook's graph representation would be dull, as the graph is regular (ie the number of attacked squares is the same at any square),  but all the other pieces, save for the pawn, would give students the visual lesson that the pieces in the center have more options.  Not only this, it appeals to an old military adage - to control the high ground - which in these cases is the center, if at all possible.  </p>
 11629: <p>Not in 3D and not super-fancy, but it should get across the point. I did not do it for the rook because as you note the rook always has 14 squares to go to (on an empty board).</p>

<p>However if you want to use this to argue for putting pieces in the center, I'd be careful, because:</p>

<ul>
<li>This is for an empty board. Lots of things can happen when other pieces are on the board.</li>
<li>The illustration for the king is probably of limited value because (unless you are Steinitz) we don't usually think of the king as an active piece.</li>
<li>For the queen and bishop (which are essentially the same, only shifted by 14, the number of rook moves) you notice that the difference between different squares (center and border) is not all that great. In my opinion a much more important factor for positioning queen and bishop centrally in a real game is that they have more options, i.e. a bishop in the center can go in four directions while a bishop in the corner has only one direction. This means that a bishop in the center can attack things in all four directions and likewise in order to block this bishop, the opponent has to block four directions. Also for transferring the bishop to other parts of a board a central position might be better.</li>
<li>For the knight there could be some value in this as it cannot be blocked by other pieces, so the situation for a knight is essentially the same as on an empty board. That's where you get the saying: "A Knight on the rim is dim" from. But as with all such rules there are plenty of exceptions.</li>
</ul>

<p><a href="https://i.stack.imgur.com/nXXZd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nXXZd.png" alt="King"></a>
<a href="https://i.stack.imgur.com/BdYxJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BdYxJ.png" alt="Knight"></a>
<a href="https://i.stack.imgur.com/QGXDC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QGXDC.png" alt="Queen"></a>
<a href="https://i.stack.imgur.com/17R2B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/17R2B.png" alt="Bishop"></a></p>
