Exercise: create the solid and surface models of the dodecahedron shown below!

 • Drawing in 2D

The radius of the circles, circumscribed about the pentagonal faces and holes of the dodecahedron are 10 and 8 units respectively.

•> Create the first pentagon with the Polygon (PG) command. First you should specify the number of sides (5), and the center of the polygon (0,0), then choose the Inscribed option (for the pentagon is inscribed in a circle), and finally specify the radius of that circle (10).

»  Of course a Polygon is not a drawing element — it's only a command simplifying the creation of PLines of a certain shape, (like Rectangle)

•> Draw the pentagonal hole also (its radius should be 8), and place the two PLines to different layers!

To be able to draw the surface model, you should first create one of its faces. The PLine elements can be converted to surfaces by the Region command, but these Regions by default can only be seen from one direction on a shaded picture. So you should the PFace element instead.

•> When you draw the polyface (PFace), you should specify the location of all it's vertexes first, and in the second phase you only have to type the numbers of the vertexes to be connected. To draw an invisible edge, type a "-" sign before the number of the first vertex of the edge! You can copy the text below to AutoCAD's command line:
PFace 10<90 10<162 10<234 10<306 10<18 8<90 8<162 8<234 8<306 8<18  -1 6 -7 2  -2 7 -8 3  -3 8 -9 4  -4 9 -10 5  -5 10 -6 1   .

•> Place this face to a third layer, and define all three elements as one block! Since the elements of a block cannot be separated, you can temporarily switch Off the two latter layers containing the smaller pentagon and the surface.

 • Drawing in 3D

•> Mirror the original (black) pentagon about it's two edges as shown on the figure!

The resulting (dashed red) faces can be rotated to their correct 3D position about their common edges with the original (black) element. In this process, their endpoints are moving along their circular paths above the dashed green lines, and they reach their correct position above point A — the intersection point of the two dashed green lines.

•> Draw a vertical Line in point A!

•> Change the coordinate system: the fist point (specifying the origin) can be point (MIDpoint) C, the second (showing the direction of the X axis) can be point B, and the third (defining the rotation angle of the XY plane) can be any point on the dashed yellow line. In this new UCS you can draw the (dashed grey) circle, along which point B moves.

•> To rotate the face to it's correct position you should rotate the pentagon containing point B with CB-CD angle (the other pentagon can be deleted).

»  The rotation angle can be shown by simply clicking on point D. But it's a special case only! When you click on a point to specify the rotation angle, the angle will be measured from the X axis to the line connecting the origin with the given point. Since in our case the X axis is on the CB line, we get the correct result. If it's not the case, you can always use the Reference option to specify a starting angle relative to which the rotation should take place:
Command: ROtate (or Modify • Rotate)
Select objects: (select the face)
Select objects: [Enter]
Specify base point: (specify C as the center of rotation)
Specify rotation angle or [Reference]: R
Specify the reference angle <0>: (specify C again)
Specify second point: (specify A or B)
Specify the new angle: (specify point D)

•> The other four bottom faces should be created using the Array command, then this half-dodecahedron should be mirrored (Mirror3d) about the plane defined by the MIDpoints of the sloping faces' top edges, and finally it should be rotated with 180°.

 • 3D model types

If you switch the third layer On, you can see that the surface model is ready.

•> To complete the solid model, you should eXplode all (ALL) blocks first, freeing the original elements.

•> Let's Freeze all layers but the first one, EXTrude ALL twelve elements, and unite them using the UNIon command.

»  When the object's own Z axis points to the UCS's -Z direction AutoCAD extrudes the object according to the UCS. To avoid this, rotate the UCS before the extrusion (UCS X 5)!

•> Repeat the extrusion with the smaller pentagons after Thawing the second layer, and SUBtract all twelve resulting solids from the previously created dodecahedron!

Strommer L. • BME Department of Architectural Representation