Go to Fun_Math Content Table Three Famous Problems Delian Problem

The first solution was given by
Archytas of Tarentum (circa 428 BC-350 BC)
.

The remarkable thing about this fact is his solution is the intersection of three surfaces of revolution,when most of his contemporary
mathematicians were dealing only with the 2D , plane geometry.

In historical perspective, he lived long before
Euclid of Alexandria (circa 325 BC-265 BC)
wrote his famous book "The Elements".

In the drawing below, the objective is to find length (AC/AB)^{1/3} for given lengths of AB & AC.

These are the steps to find such line.

(1) Three surfaces of revolution meets at point P.

(2) Draw a line PM perpendicular to plane ABC.

(3) Point M will be on the circle ABC.

(4) Then AC/AP = AP/AM = AM/AB

(5) Therefore AC/AB = (AM/AB)^{3}

if AB = 1, and AC = 2, then AM = (2)^{1/3}

For the geometrical arguments regarding statements (3) & (4),refer to the ref.1 by Heath,

Here let us use analytical geometry to derive this.

Analytically three surfaces of revolution can be written as follows:

(1) the right cone : x^{2} + y^{2} + z^{2} = (a/b)^{2}x^{2}

(2) the cylinder : x^{2} + y^{2} = a x

(3) the torus : x^{2} + y^{2} + z^{2} = a {x^{2} + y^{2}}^{1/2}

Combining (1) & (2), we have x^{2} + y^{2} + z^{2} = {x^{2} + y^{2}}^{2} / b^{2}

From this and (3), we obtain

a / {x^{2} + y^{2} + z^{2}}^{1/2} = {x^{2} + y^{2} + z^{2}}^{1/2} / {x^{2} + y^{2}}^{1/2} = {x^{2} + y^{2}}^{1/2} / b

or, AC / AP = AP / AM = AM / AB

*********** Archytas_Delian_model.dwg** *********
********** Archytas_Delian_result.dwg** ********
**To create this drawing : **
** Load Archytas_Delian.lsp (load "Archytas_Delian")**

Then from command line, type **Archytas_delian_model ** for the left

Similarly from command line, type **Archytas_delian ** for the right.

Numerical Check of the result

Let us check the numerical value of line length AM,which is {x^{2} + y^{2}}^{1/2}.

Substituting (2) into (3), we obtain

x^{2} + y^{2} + z^{2} = a ( a x )^{1/2} ------------------------- (4)

Equating the right hand side of (1) and (4), x can now be determined as x = {b^{4}/ a }^{1/3} ---------- (5)

Substituting (5) into the right hand side of (2), we obtain

x^{2} + y^{2} = a x = {a^{2} b^{4}}^{1/3}

Therefore, AM = {x^{2} + y^{2}}^{1/2} = {ab^{2}}^{1/3}

The value for y & z can also be computed from the following result.

y^{2} = {a^{2} b^{4}}^{1/3} - {b^{8} / a^{2}}^{1/3}

z^{2} = {a^{4} b^{2}}^{1/3} - {a^{2} b^{4}}^{1/3}

If we let a = AC = 2, and b = AB = 1, then

AM = (2)^{1/3} and AP = (4)^{1/3}

and x = (1/2)^{1/3} ; y^{2} = (4)^{1/3} - (1/4)^{1/3} ; z^{2} = (16)^{1/3} - (4)^{1/3}

or, x = 0.7937005 , y = 0.9784889 , z = 0.9656298 , which is the 3-D coordinate value of the point "P".

the equation for this cylinder: x

******************************

Then from command line, type

The resulting semicircle is vertical to ABC plane. Rotate this semi-circle around z-axis 90 degrees.

The result is a quarter of a torus with zero inner diameter.

the equation for this torus: x

******************************

Then from command line, type

This will create a quarter of a cone.

the equation for this cone: x

.

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Then from command line, type

******************************

Then from command line, type

Then split viewports into 4, using VPORT command. Change view point in each view port

******************************

Then from command line, type

Then use INTERSECT command to modify surfaces.

Split into 4 viewports, and change view directions.

*********************************

Explode the surface objects. Then the boundary lines become independent lines.
Locate the point "P" using "END" osnap command.

*********************************** explode_find_point_p.dwg** *********************************

Using LIST command, the following value will be displayed on the TEXT screen.

POINT Layer: "0"

Space: Model space

Handle = 663

at point, X=0.79370053 Y=0.97848890 Z=0.96562987

************ point_of_intersection.dwg** **********

1. Heath, Sir L. Thomas:"A History of Greek Mathematics", VOL. 1, From Thales to Euclid., Dover.

Go to Fun_Math Content Table Three Famous Problems Delian Problem

All questions/suggestions should be sent to Takaya Iwamoto

Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved. .