Ad Quadratum Construction and Study of the Regular Polyhedra
Ad Quadratum Construction and Study of the Regular Polyhedra
Ad Quadratum Construction and Study of the Regular Polyhedra
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35<br />
Ratio <strong>of</strong> Intersphere to Circumsphere Radii<br />
Besides <strong>the</strong> insphere just described, ano<strong>the</strong>r sphere can be considered in relation to <strong>the</strong><br />
platonic polyhedra -–<strong>the</strong> intermediate sphere or intersphere 13 . This sphere is defined by<br />
<strong>the</strong> midpoints <strong>of</strong> each edge <strong>of</strong> a platonic polyhedron, marking <strong>the</strong> intercept <strong>of</strong><br />
interpenetrating duals.<br />
The radius <strong>of</strong> this intersphere can be readily calculated for each polyhedron.<br />
Let ri be <strong>the</strong> radius <strong>of</strong> <strong>the</strong> intersphere.<br />
Referring to fig. 5, which applies to all <strong>the</strong> regular convex polyhedra, <strong>and</strong> represents one<br />
<strong>of</strong> <strong>the</strong> lateral faces <strong>of</strong> <strong>the</strong> pyramids making up each polyhedron, it is seen that <strong>the</strong> radius<br />
C ˆ i<br />
C ˆ i<br />
<strong>of</strong> <strong>the</strong> intersphere will be CJ <strong>and</strong> in all cases, CJ ri R cos ; where here is half<br />
2 2<br />
<strong>the</strong> internal angle <strong>of</strong> a particular polyhedron.<br />
From <strong>the</strong> identity cos A 1<br />
1cos A<br />
we <strong>the</strong>refore obtain for each polyhedron, in<br />
2 2<br />
turn:<br />
Tetrahedron:<br />
Cube:<br />
Octahedron:<br />
ri<br />
Tˆ<br />
i 1 1 1<br />
cos 1<br />
<br />
R 2 2 3 3<br />
ri<br />
Cˆ<br />
i 1 1 <br />
cos 1<br />
<br />
R 2 2 3 <br />
ri O ˆ i<br />
cos<br />
R 2<br />
but sin ˆ<br />
O i 1 cos ˆ<br />
O i 0<br />
13 Kappraff, Jay: Connection, McGraw-Hill, Inc., New York, 1991, p. 266.<br />
2<br />
3<br />
3<br />
3