Construction of Regular Icosahedron within Given Sphere

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Theorem

In the words of Euclid:

To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the icosahedron is the irrational straight line called minor.

(The Elements: Book $\text{XIII}$: Proposition $16$)


Porism

In the words of Euclid:

From this it is manifest that the square on the diameter of the sphere is five times the square on the radius of the circle from which the icosahedron has been described, and that the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle.

(The Elements: Book $\text{XIII}$: Proposition $16$ : Porism)


Proof

Euclid-XIII-16.png

Let $AB$ be the diameter of the given sphere.

Let $AB$ be cut at $C$ where $AC = 4 \cdot CB$.

Let $ADB$ be a semicircle on the diameter $AB$.

Let $CD$ be drawn from $C$ perpendicular to $AB$.

Let $DB$ be joined.

Let the circle $EFGHK$ be set out whose radius equals $DB$.

From Proposition $11$ of Book $\text{IV} $: Inscribing Regular Pentagon in Circle:

let the regular pentagon $EFGHK$ be inscribed within the circle $EFGHK$.

Let the arcs $EF, FG, GH, HK, KE$ be bisected at the points $L, M, N, O, P$.

Let $LM, MN, NO, OP, PL, EP$ be joined.

It follows that the pentagon $LMNOP$ is regular.

Also, the straight line $EP$ is the side of a regular decagon.

From the points $E, F, G, H, K$ let the straight lines $EQ, FR, GS, HT, KU$ be constructed perpendicular to the plane containing the circle $EFGHK$.

Let each of $EQ, FR, GS, HT, KU$ be equal to the radius of circle $EFGHK$.

Let $QR, RS, ST, TU, UQ, QL, LR, RM, MS, SN, NT, TO, OU, UP, PQ$ be joined.

We have that $EQ$ and $KU$ are both perpendicular to the same plane.

From Proposition $6$ of Book $\text{XI} $: Two Lines Perpendicular to Same Plane are Parallel:

$EQ \parallel KU$

But we also have that $EQ = KU$.

So from Proposition $33$ of Book $\text{I} $: Lines Joining Equal and Parallel Straight Lines are Parallel:

$QU = EK$

and:

$QU \parallel EK$

But $EK$ is the side of the regular pentagon $EFGHK$ which is inscribed within the circle $EFGHK$.

Therefore $QU$ is also the side of the regular pentagon $EFGHK$ which is inscribed within the circle $EFGHK$.

For the same reason:

each of the straight lines $QR, RS, ST, TU$ are the sides of the regular pentagon $EFGHK$ which is inscribed within the circle $EFGHK$.

Therefore $QRSTU$ is itself a regular pentagon which can be inscribed within a circle equal to the circle $EFGHK$.

We have that $\angle QEP$ is a right angle.

Therefore from Proposition $47$ of Book $\text{I} $: Pythagoras's Theorem:

$QE^2 + EP^2 = QP^2$

We have that $QE$ equals to the radius of circle $EFGHK$.

From Porism to Proposition $15$ of Book $\text{VI} $: Inscribing Regular Hexagon in Circle:

$QE$ equals the side of a regular hexagon which could be inscribed within the circle $EFGHK$.

We also have that $EP$ equals the side of a regular decagon which could be inscribed within the circle $EFGHK$.

Therefore from Proposition $10$ of Book $\text{XIII} $: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in same Circle:

$QP$ equals the side of the pentagon $EFGHK$.

For the same reason:

$PU$ equals the side of the pentagon $EFGHK$.

But also:

$QU$ equals the side of the pentagon $EFGHK$.

Therefore $\triangle QPU$ is equilateral.

For the same reason:

$\triangle QLR, \triangle RMS, \triangle SNT, \triangle TOU$ are all equilateral triangles.

We have that:

$QL$ and $LP$ have been proved to equal the side of the pentagon $EFGHK$.

We also have that $LP$ equal the side of the pentagon $EFGHK$.

Therefore $\triangle QLP$ is equilateral.

For the same reason:

$\triangle LRM, \triangle MSN, \triangle NTO, \triangle OUP$ are all equilateral triangles.

Let $V$ be the center of the circle $EFGHK$.

Let $VZ$ be constructed perpendicular to the plane containing the circle $EFGHK$.

Let $VZ$ be produced in the other direction as $VX$.

Let $VW$ be cut off $VZ$ equal to the side of a regular hexagon which could be inscribed within the circle $EFGHK$.

Let each of $VX$ and $WZ$ be made equal to the side of a regular decagon which could be inscribed within the circle $EFGHK$.

Let $Q, QW, UZ, EV, LV, LX, XM$ be joined.


We have that $VW$ and $QE$ are both perpendicular to the same plane.

From Proposition $6$ of Book $\text{XI} $: Two Lines Perpendicular to Same Plane are Parallel:

$VW \parallel QE$

But we also have that $EQ = KU$.

So from Proposition $33$ of Book $\text{I} $: Lines Joining Equal and Parallel Straight Lines are Parallel:

$EV = QW$

and:

$EV \parallel QW$

But $EV$ equals the side of a regular hexagon which could be inscribed within the circle $EFGHK$.

Therefore $QW$ also equals the side of a regular hexagon which could be inscribed within the circle $EFGHK$.

We have that $\angle QWZ$ is a right angle.

Therefore from Proposition $47$ of Book $\text{I} $: Pythagoras's Theorem:

$QW^2 + WZ^2 = QZ^2$

Therefore from Proposition $10$ of Book $\text{XIII} $: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in same Circle:

$QZ$ equals the side of the pentagon $EFGHK$.

We have that $WU$ equals the side of a regular hexagon which could be inscribed within the circle $EFGHK$.

But $WZ$ equals the side of a regular decagon which could be inscribed within the circle $EFGHK$.

Also $\angle UWZ$ is a right angle.

So for the same reason as before:

$UZ$ equals the side of the pentagon $EFGHK$.

But $QU$ also equals the side of the pentagon $EFGHK$.

Therefore $\triangle QUZ$ is equilateral.

For the same reason:

each of the remaining triangles of which the straight lines $QR, RS, ST, TU$ are the bases, and the point $Z$ the apex, is equilateral.

We have that:

$VL$ equals the side of a regular hexagon which could be inscribed within the circle $EFGHK$

and:

$VX$ equals the side of a regular decagon which could be inscribed within the circle $EFGHK$.

We also have that $\angle LVX$ is a right angle.

So for the same reason as before:

$LX$ equals the side of the pentagon $EFGHK$.

Suppose $MV$ were joined.

We have that:

$MV$ equals the side of a regular hexagon which could be inscribed within the circle $EFGHK$

and:

$MX$ equals the side of a regular decagon which could be inscribed within the circle $EFGHK$.

We also have that $\angle MVX$ is a right angle.

So for the same reason as before:

$MX$ equals the side of the pentagon $EFGHK$.

Therefore $\triangle LMX$ is equilateral.

For the same reason:

each of the remaining triangles of which the straight lines $MN, NO, OP, PL$ are the bases, and the point $X$ the apex, is equilateral.

Thus we have created an icosahedron which is contained by $20$ equilateral triangles.

$\Box$


Next it is to be demonstrated that it can be inscribed in the given sphere.

We have that:

$VW$ equals the side of a regular hexagon which could be inscribed within the circle $EFGHK$.

and:

$WZ$ equals the side of a regular decagon which could be inscribed within the circle $EFGHK$.

Therefore from Proposition $9$ of Book $\text{XIII} $: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio:

$VZ$ has been cut in extreme and mean ratio whose greater segment is $VW$.

Therefore:

$ZV : VW = VW : WZ$

But:

$VW = VE$

and:

$WZ = VX$

Therefore:

$ZV : VE = EV : VX$

We have that $\angle ZVE$ and $\angle EVX$ are right angles.

We have that $\triangle XEZ$ and $\triangle VEZ$ are similar.

Therefore if $EZ$ is joined, $\angle XEZ$ will be a right angle.

For the same reason:

$ZV : VW = VW : WZ$

and:

$ZW = XW$

and:

$VW = WQ$

Therefore:

$XW : WQ = QW : WZ$

And for the same reason again:

if $QX$ is joined, $\angle Q$ will be a right angle.

Therefore from Proposition $31$ of Book $\text{III} $: Relative Sizes of Angles in Segments:

the semicircle described on $XZ$ also passes through $Q$.

Let $XZ$ remain fixed, and let that semicircle be carried around and restored to the same position from which it began.

It will also pass through $Q$ and the remaining vertices of the icosahedron.

Thus the icosahedron has been inscribed in a sphere.


Let $VW$ be bisected at $A'$.

We have that $VZ$ has been cut in extreme and mean ratio at $W$, where the lesser segment is $ZW$.

Therefore from Proposition $3$ of Book $\text{XIII} $: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio:

$\left({ZW + WA'}\right)^2 = 5 \cdot WA'^2$

Therefore:

$ZA'^2 = 5 \cdot A'W^2$

Also:

$ZX = 2 \cdot ZA$

and:

$VW = 2 \cdot A'W$

Therefore:

$ZX^2 = 5 \cdot WV^2$

We have that:

$AC = 4 \cdot CB$

Therefore:

$AB = 5 \cdot BC$

But from:

Proposition $8$ of Book $\text{VI} $: Perpendicular in Right-Angled Triangle makes two Similar Triangles

and:

Book $\text{V}$ Definition $9$: Duplicate Ratio

it follows that:

$AB : BC = AB^2 : BD^2$

Therefore:

$AB^2 = 5 \cdot BD^2$

But:

$ZX = 5 \cdot VW$

and:

$DB = VW$

because each of them equals the radius of the circle $EFGHK$.

Therefore:

$AB = XZ$

Also, $AB$ is the diameter of the given sphere.

Therefore $XZ$ is also the diameter of the given sphere.

Therefore the icosahedron has been be inscribed in the given sphere.

$\Box$


It remains to be demonstrated that the side of the icosahedron is the irrational straight line called minor.

We have that the diameter $AB$ of the given sphere is rational.

We have:

$AB^2 = 5 \cdot EV^2$

where $EV$ is the radius of the circle $EFGHK$.

Therefore $EV$ is also rational.

Therefore the diameter of the circle $EFGHK$ is rational.

But from Proposition $11$ of Book $\text{XIII} $: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor:

the side $EF$ of the regular pentagon $EFGHK$ is minor.

But the side $EF$ of the regular pentagon $EFGHK$ is also the side of the icosahedron.

Hence the result.

$\blacksquare$


Historical Note

This proof is Proposition $16$ of Book $\text{XIII}$ of Euclid's The Elements.
Some sources credit this proof to Theaetetus of Athens.


Sources