Rediscovering Nomograms in a Digital Age

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Rediscovering Nomograms in a Digital Age

• Chuck Pheatt and Jorge Ballester, Emporia State
  University , Emporia , KS

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This is a Nomogram
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What is a Nomogram?

• The graphical representation of an equation,
  usually of three variables f(u, v, w) 0, by
  means of graphical scales (not necessarily
  straight), arranged in such a manner that any
  straight line, called an index line, cuts the
  scales in values of u, v, and w satisfying the
  equation.
• From the American Meteorological Society
  glossary.

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Motivation

• Historical interest.
• Connection to old computation, e.g. slide rules.
• Building number sense.
• The purpose of computing is insight not
  numbers. Hamming
• Assess relevance in calculator age.
• Automated generation (perhaps AI)
• Aftermath of library purges books and associated
  content are lost (e.g. base 12).

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Computational Aids

• Graphical aids
• Slate and coal
• Calculators (people)
• Analog computers
• Digital computers
• Tables (log, tabulated functions, interpolation)
• xy graphs (linear, semi-log, log-log)
• Mechanical calculators (Frieden)
• Electronic slide rule calculators

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Graphical/Analog Devices

• Slide rule
• Nomogram
• Sector
• Abacus
• Navigation/flight computer

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Slide Rule Basics

• Scales
• Addition of lengths
• Multiplication by addition of log scales
• Lookup scales (e.g. trig, exp, sqr, sqrt)
  nomogram of class 2, genus 0
• Exotic arrangements (circular, spiral)

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Slide Rule Multiplication

• Post Versalog C D scales

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Make Your Own Slide Rule

• Brians slide rule http//www.sphere.bc.ca/test/bu
  ild.html

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Definition of Class

• Class is the number of variables represented by
  the nomogram.

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Circle Area Calculator

• Scale equation

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Class 1 Nomogram
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Class 2 Nomograms

• Two variables, vf(u).
• Generate y-coordinates using scale equations
  f1(u) and f2 (v).
• Also called a conversion scale.
• Can be represented along one straight line but
  can be curved for convenience.
• Scales are not necessarily uniformly spaced!

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Simple Class 2 Nomogram

• Graphical version of table lookup.
• Inch to millimeter conversion.
• f1(u)? u
• f2(v) ? v/25.4

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One Very Long Nomogram

• A Graphic Table Combining Logarithms and
  Anti-logarithms, A. Lacroix and C. L. Ragot
  (1925, 5th printing 1941).
• Base 10, six-decimal places.
• 40 pages X 25 strips per page X 4.5 inches per
  strip 450 feet

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Classes and Genera

• System due to dOcagne Traite de Nomographie
  (1921).
• For class 3, the three variable axes must be
  separated. Placement now depends on the
  functional relationship.
• Functional relationship defines the locus and
  scale.
• Genus specifies the number of curved axes.

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Class 3, Genus 0

• Three variables, zero curved scales.
• Example three line addition nomogram
  Z X Y

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Addition Nomogram
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Geometric Basis
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How It Works

• From geometry,
• Rearranging,
• Constraint,

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Lines and Determinants

• In general, for three collinear points,
• Rearranging into determinant form,

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Determinants and Nomograms

• This,
• may be rewritten as,

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Allcock Jones Fig 7 (The Epiphany Graph)
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BMI
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BMI Surface Plot
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BMI Contour Plot
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BMI Nomogram Development

• BMI (W / H2) 703
• re-written into the form f1(h) f2(R) f3(H)
  0
• log(W) log(BMI) - 2log(H) log(703) 0
• f1(W) log(W)
• f2(BMI) log(BMI) log(703)
• f3(H) - 2log(H)

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BMI Nomogram Development

• Substituting our equation into the standard form
• -D f1(W) log(W) 1
• 0 f2(BMI) log(BMI) log(703) 1
• D f3(H) - 2log(H) 1

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BMI Nomogram Development

• The four functional moduli (D1, D3, U1, U3), with
  the constraint D1 U3 D3 U1
• -D1 f1(h) U1 log(W)
  1
• 0 f2(R) (log(BMI) log(703)) (U1
  U3)/(U1 U3) 1
• D3 f3(H) U3 (- 2log(H) )
  1
• Setting D1 1.2, D3 1.8, U1 1.2, U3 1.8.

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The Finished Product
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With Revised BMI Scale
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BMI Nomogram
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Projectile Range
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Projectile Range
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Class 3, Genus 1

• Genus 1 ? one curved axis
• During the depression when gag writers on the
  radio were dismissed without notice, each was
  given W weeks pay (0 to 5), if he/she had C
  children (0 to 8), and had been employed for Y
  years (0 to 4) in accordance with
  W/Y 1 C / WY where W is the number of
  weeks pay. C is the number of children. Y is
  the number of years employed
• Reference Davis, Nomography of Empirical
  Equations

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Dismissal Nomogram
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How far can you go?

• General formula for class 3, genus 3,

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Class 3, Genus 3
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Class 3, Genus 3
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What does this have to do with biology?
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What does nomography have to do with learning
mathematics?

• Construction exercise
• Calculation tool
• Historical significance

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Good Things bout Nomograms

• Robust
• Visual
• Numerical
• Simplifying
• Play what if easily
• Approximating
• 3-variable relationship
• Easier to read than 3d surface plots
• Interesting in a nerdy way
• Insight through calculation (Hamming quote)
• Visualize the range of possible solutions

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Bad Things bout Nomograms

• Approximate
• Paperful
• A pain to make!
• No more draftsmen
• Limited applicability
• Lack of framework for creation (We used Excel
  POVray)

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Books

• Allcock and Jones, The Nomogram, 4th edition
  (1950).
• Davis, Empirical Equations and Nomography (1943).
• Kraitchik, Alignment Charts Construction and Use
  (1944).
• Many more

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Web Resources

• Project rho (gaming) http//www.projectrho.com/nom
  ogram/
• Online tool http//www.ece.rochester.edu/jones/No
  moDevel/