Multiply and Divide Geometrically Like Mother Nature

This method of multiplication and division was used by Descartes and is from Euclid's "Elements", Book VI, Proposition 12. It is based on similar triangles. It may very well be the way Mother Nature accomplishes Multiplication and Division! One imagines that Nature might be able to create straight lines through the emission of rapid vibrations through tightly packed particles or molecules. See the article Center a Circle and think how it might work in reverse in order to accomplish just this requirement. However, this is only a theory, a possibility; Science knows that Nature accomplishes mathematical wonders, such as phyllotaxis, and growth patterns very much like fractal iterative patterns, but is still debating just how She accomplishes that! It's well worth thinking about and devising experiments and empirical evidence for proof.


  • Become acquainted with the image of the basic concept:

The tutorial

  1. Look at these similar triangles, and therefore the proportion {DG}/{DH}={DE}/{DF}. You can use it to perform multiplication and division. Open a new workbook in Excel and copy the drawing.
  2. To multiply x times y, make horizontal line DH of length 1, extend DF of length x from DH and raise DG of length y at an angle above horizontal DF.  Draw HG and construct a line through F parallel to HG.  Let it intersect DG at E. Then DE will have length xy.
  3. To divide y by x, make DH of length 1, DF of length x and DE of length y.  Draw EF and construct a line through H parallel to EF.  Let it intersect DE at G. Then DG will have length y/x.
  4. Suppose one stem or leaf underlying another, in its shadow. Would this possibly be a way of keeping time and "knowing when to move aside" to attain better light, directly, for the lower leaf or stem?
  5. Suppose crossing roots (which they do) and suppose some sensitivity to one another -- could this be a way plants perform math and send vital nutrients on a timely basis up the plants? After all, the roots are in darkness, how do they know what time it is or calculate the proportion of a given chemical admixture to send?
  6. Suppose neurons branching at various angles in the brain (which they do) -- could this be a way of computation of p/n = A.E.N. (Almost Any Number)? That is, almost any number may be expressed as a quotient of two other numbers, e.g. 36/2 = 18 and 625/256 = 2.44140625, or 5^4 / 4^4 or 5/4^(1/(5/4 - 1)). See the articles Start Working with Continued Fractions and Solve aB=a^B in Neutral Operations Using Algebra where it discusses E = mc^n as n approaches 2. Is it possible to "see yesterday" in the memory by seeing slower than stopped Time at light speed squared? Is the "Past" on the opposite side of all electrons facing me, and the "Future" spinning around from that opposed position to greet me too? That would make the immediate Past very much like the immediate Future, resulting in a fairly stable Present. And geometrically, all the rays from all the particles coming through in vibrations would be fairly constantly multiplied and divided as well, so long as one is fairly still, or in a relatively stable environment. Call this "The Supposition About Neurons and Neutrons" if you like.
  7. Descartes also used the next proposition, VI.13, to take square roots geometrically.

Stay Curious

  1. If it can be done geometrically, then can Mother Nature perform it within reasonable tolerances? That is, can She get reasonable estimates of the square root or of any root of a number? One surmises "any root" from supposing an iterative process (that did not occur to Euclid, Descartes or Newton-Raphson apparently).
  2. Final Image:

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.


  • a*b = a/b = c has just 1 answer, 1, because:
  • if and when ab/a = a/ab
  • b = 1/b and b must = 1. If it = 0, then 0 is equated to ∞ (infinity) because ∞ = 1/0 or 1/x as x approaches 0, i.e. Nothingness everywhere -- a possible primordial state of the Universe in some theories. This is gathered from the tangent y/x of 90 degrees (the y-axis) as x approaches 0; in order for the x and y axes to be perpendicular, INF * 0 = -1, since the tangent y/x of 0 degrees (the x-axis) = 0. The axes are not Undefined; hardly, they exist, albeit as approximations, but as an ideal, this is the truth of their relation. And that does not imply Nothingness Everywhere to many decent math students.
  • This is interesting because it surrenders up base 2, comprised of 0 and 1. Or Nothing and Unity. Please see Related Articles for an interesting article on creating -1 and 1 from 2-3 "differently sized" zeroes (or spaces, or Space-Times) and the Null Set.

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