To construct a dodecahedron, and to enclose (it) in a sphere, like the aforementioned figures, and to show that the side of the dodecahedron is that irrational (straight line) called an apotome. * Let two planes of the aforementioned cube [Prop. 13.15], $ABCD$ and $CBEF$, (which are) at right angles to one another, be laid out. * And let the sides $AB$, $BC$, $CD$, $DA$, $EF$, $EB$, and $FC$ have each been cut in half at points $G$, $H$, $K$, $L$, $M$, $N$, and $O$ (respectively). * And let $GK$, $HL$, $MH$, and $NO$ have been joined. * And let $NP$, $PO$, and $HQ$ have each been cut in extreme and mean ratio at points $R$, $S$, and $T$ (respectively). * And let their greater pieces be $RP$, $PS$, and $TQ$ (respectively). * And let $RU$, $SV$, and $TW$ have been set up on the exterior side of the cube, at points $R$, $S$, and $T$ (respectively), at right angles to the planes of the cube. * And let them be made equal to $RP$, $PS$, and $TQ$. * And let $UB$, $BW$, $WC$, $CV$, and $VU$ have been joined. * I say that the pentagon $UBWCV$ is equilateral, and in one plane, and, further, equiangular. * ... Thus, if we make the same construction on each of the twelve sides of the cube then some solid figure contained by twelve equilateral and equiangular pentagons will have been constructed, which is called a dodecahedron. * ... So, I say that the side of the dodecahedron is that irrational straight line called an apotome.
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Proofs: 1