Determine a Line = to Square Root of 3 Geometrically

For a cube, the long internal diagonal from bottom to top diagonally cross-corner to cross-corner = side times the square root of 3, which is related to the formula for the diagonal for a square, side times the square root of 2. These follow from the Pythagorean Theorem: the latter due to the fact that 1^2 + 1^2 = sqrt(2)^2, and for the cube, 1^2 + sqrt(2)^2 = sqrt(3)^2, that is, the height squared plus the floor diagonal squared = the long top-to-bottom cross-corner diagonal squared. The square root of 3 was discovered geometrically upon extrapolation from BOOK I PROPOSITION 1 of Euclid's "Elements".

Steps

The Tutorial

  1. Get to know the image you'll be creating.
  2. Make a given finite blue horizontal line of unit length = 1, and treating each endpoint as the center of a radius, make two overlapping circles.
  3. Connect the endpoints of the original line (radius) from either side with the intersection point of the two circles. Both top and bottom, with straight lines will form two equilateral triangles, one atop the other, the bottom one an inverted mirror image of the top triangle. All the radii are equal and all sides being equal, these are proven equilateral triangles.
  4. Drop the connecting perpendicular between the top intersection point of the two circles and the bottom intersection point of the two circles. The length of this line equals the square root of 3.
  5. Do the math. Where the perpendicular cuts the original given unit line to the line's left (or right) endpoint is a distance of .5 -- let us call this distance a. a^2 = .25. The hypotenuse has a length of 1; let us call the hypotenuse c and c^2 = 1. c^2 - a^2 = b^2 = 1 - .25 = 3/4 and the square root of this is sqrt(3)/2 and equals 1/2 the dropped perpendicular between the intersection points, top and bottom, of the two circles. Therefore twice this distance, or the measure of the full perpendicular between the circle's intersection points, equals sqrt(3)/2 * 2 which = the square root of 3 ... the very distance which was sought to be determined geometrically.

Explanatory Charts, Diagrams, Photos

  1. The black line equals the square root of 3 relative to the radius of 1 between 0 and +1 on the x axis. Sqrt(3) = 1.73205080756888 and we can see the black line is about 2*.85 or 1.7 units in length, roughly.

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
    • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.



Tips

  • There is often more to Euclid than is obvious in the proof, if one does a little extra thinking.
    • For example, in BOOK I, PROPOSITION 2 of the "Elements", what is not obvious is that the baby step taken forward by the line transferred to an arbitrary point has actually been rotated 180 degrees -- heel to toe. If one instead of the proof draws a straight line from point c to point a and then creates a circle equal in radius to the length of the old line, one will have transferred the line (with 360 degrees of freedom) to the arbitrary point. However, one sacrifices, the triple-check which the equilateral triangle method used in Euclid's proof provides.
  • Since side s^3 = a cube, and sqrt(3) * sqrt(3) = 3, it follows that we may say that s^(sqrt(3)^2) = a cube. In other words, by multiplying two cross diagonals together in the exponent we obtain a cube.

Related Articles

References

  • The source workbook used for this article is "SQRT 3 WORKBK.xlsx"
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