Architectural Informatics 2 |
Dodecahedron (3D) |
Digital Representation |

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

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 **PLine**s of a certain shape, (like **Rectangle**)

•> Draw the pentagonal hole also (its radius should be 8), and place the two **PLine**s 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 **Region**s 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.

•> 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 __M__odify • __R__otate*)*

`Select objects: `*(select the face)*

`Select objects: [Enter]`

`Specify base point: `
*(specify C as the center of rotation)*

•> 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°.

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 *Thaw*ing the second layer, and **SUBtract** all twelve resulting solids from the previously created dodecahedron!

2006. Strommer L.
** • BME Department of Architectural Representation**