signatures

The Doctrine of Signatures
by Stephen Wirtz
Salmen High School Early Alchemists believed that by observing nature ( such as the flight of birds or the ferocity or lightning ) they could find the qualities inherent within the structure of the elements. This was called the Doctrine of Signatures and while not a productive approach for observational science, it did later prove useful for speculative psychology ( refer to the writings of Carl Jung on Alchemy ).
However, modern science does use what may be called a variant of the Doctrine of Signatures when it utilizes graphs or other visual representations to summarize and organize data. The Hertzsprung-Russell Diagram which plots the luminosity of stars on the vertical axis and their spectral class ( their color ) on the horizontal axis is an example of this type of approach.

The line through the center of of the diagram is known as the Main Sequence of stars. It is a type of signature.
It may be usefull to utilize the position function, which is height above ground = ( acceleration of gravity/2 )(time squared) + (initial velocity)(time) + initial height or more compactly y = Ax^2 + Bx + C to compare the gravitational signatures of various planetary bodies.

EQUIPMENT:
1. Graphing calculator ( if you have a Casio with a dynamic mode it may be used to introduce each of the following three cases---it adds life to the presentation but is strictly optional ).
2. Graph link (optional ) I used it to illustrate the graphs in this lesson plan and your students will enjoy using it.
3. Graph paper ( optional ) If you choose to have them manually complete the square, it would help to plot results on graph paper.

Core equation:
y = Ax^2 + Bx + C where A = ( Acceleration of gravity )/2; B = initial velocity; C = initial height above ground.

General approach:

I used the TI-82 calculator. Reiterations of a single equation may be expressed as y = L(n)x^2 + Bx + C, where L(n) is a list of coefficients which you wish to graph ( of course, B and C can also be expressed as lists ).
Three cases are considered:
Case 1:
A and B are held constant ( for a given planet ) while C is varied by increments of your or your students choosing. I used the following equations:
Earth ( -32ft/s^2 for acceleration of gravity )                                           Mars ( -12ft/s^2 for acc. of gravity )
Set B = 1 and C ( initial ) = 1                                                                  Set B = 1 and C ( initial ) = 1
y = -16x^2 + x + 1                                                                                  y = -6x^2 + x + 1
y = -16x^2 + x + 2                                                                                  y = -6x^2 + x + 2
y = -16x^2 + x + 3                                                                                  y = -6x^2 + x + 3
y = -16x^2 + x + 4                                                                                  y = -6x^2 + x + 4

Plots of vertexes of each parabola
Use the statistical plots on your calculators if you wish to get this result
Earth Mars Notice that although the Mars plot is slightly higher for each corresponding vertex ( why would you expect this?---lower gravity ) the main visual feature is the offset of the plots to the right. Having your students complete the square on some of the above equations ( say, the first equation for Earth and the first equation for Mars), will explain why the gravitational difference expresses itself as a horizontal shift as well as a vertical shift.

Case 2:
A is held constant ( for a given planet ) and C is held constant. Coefficient B becomes a variable. If you have a Casio, the dynamic mode gives a good illustration of what happens for the following equations ( be sure to empty as much memory as possible before using this mode ). Ask the students if they can identify an equation describing the translation of the vertexes.
Suggested equations:
Earth:                                                                                                              Mars:
y = -16x^2 -9x + 1                                                                                          y = -6x^2 -9x + 1
y = -16x^2 -6x + 1                                                                                          y = -6x^2 -6x + 1
y = -16x^2 -3x + 1                                                                                          y = -6x^2 -3x + 1
y = -16x^2 +1                                                                                                 y = -6x^2 + 1
y = -16x^2 +3x + 1                                                                                         y = -6x^2 + 3x + 1
y = -16x^2 +6x + 1                                                                                         y = -9x^2 + 6x + 1
y = -16x^2 +9x + 1                                                                                         y = -9x^2 + 9x + 1
Plots of parabolas for Earth                                                Plots of parabolas for Mars

Be sure to keep the scale constant in your viewing window to accurately compare the results. Set your max and min values for x and y scale to just bracket the values given by the vertexes for best results. What is varying here is initial velocity. Notice how higher gravity " pulls in " the graph. This gives a very good " feel " for the gravitational signature of each planet under the given conditions.
If your students have been doing a lot of curve fitting to scatter plots, getting a scatter plot of the coordinates of the vertexes for each situation may suggest itself to them. They can use the calculator to find the maximums of each equation and enter the results in the lists. A quadratic regression will probably be the obvious first choice.
Scatter plot for vertexes of Earth parabolas

regression yields y = 16.2x^2 + 1

Scatter plot for vertexes of Mars parabolas

regression yields y = 6x^2 + 1

I used "zoomstat" when I plotted these points. This means that the calculator did not keep the window settings equal in order to fill the entire screen. Using another setting for your window and holding this setting constant for both scatter plots will show the differences between the two scatter plots more prominently. You may suggest that your students try different screen settings on their own. The regressions suggest families of parabolas whose vertexes fall on y = -Ax^2 + C, where A is one half of the acceleration of gravity for the given planet and C is the initial height.
To verify this result analytically, you can point out ( or have them complete the square ) that the vertex of a parabola has the coordinates ( -B/(2A), C - (B^2)/(4A)). That is, when x = -B/(2A) then y = C - (B^2)/(4A).
x = -B/(2A) => B = -2Ax and y = C - (B^2)/(4A) => B = (-4A(y - C))^(1/2).
Equating the two underlined expressions above yields y = -Ax^2 + C. This confirms the expressions yielded by quadratic regression.

Case 3:
For case 3, A is the variable while B and C are held constant. Since A represents gravitational acceleration divided by two, case three will use more planets to get a representative sample of parabolas ( I have left out Pluto because the gravity is so low it is difficult to fit its parabola on the view screen of the graphing calculator without cramping the rest of the parabolas together ( have them try it by using -0.8 for A in the equation representing Pluto )). I have also used a value of 30 for B. This allows the vertexes to spread out more when graphing ( if you have them use a smaller value for B such as 1 or 10 they can observe this for themselves. This helps them to understand that when using this approach for analyzing the behavior of graphs, the values of constants can be fixed on the basis of getting a good visual inspection of the graphs).
Suggested equations:
y = -6x^2 + 30x +1 Mercury/Mars
y = -14x^2 + 30x + 1 Venus
y = -16x^2 + 30x + 1 Earth
y = -37.5x^2 + 30x + 1 Jupiter
y = -14.5x^2 + 30x + 1 Saturn/Neptune

Ask the students to relate the heights of the vertexes to the gravitational pull of each planet. Does the translation of the vertexes seem to be following any given graph? Have the students use their calculators to find the coordinates of the vertexes of each parabola and plot these coordinates using the statistical plotting mode of the calculator.

What graph do the coordinates of the vertexes seem to be following? Since the points seem to be linear, they may try doing a linear regression using the statistical features of the calculator. My regression, with rounding, yielded y = 15x + 1. Some students may try using the median median line to get a fit. If so, this would be a good place to explore residuals using the stat mode of the calculator. Generally, the guidebook which comes with the calculator will have a short lesson on how to do this.
To verify these results analytically, the students can use the same technique as was used in Case 2. That is, the vertex of a parabola can be given by ( -B/(2A), C - ( B^2 )/( 4A ) ). This can be seen by completing the square. This means that when y = C - ( B^2 )/( 4A ) then x = -B/( 2A ). The first equation yields A = ( - B^2 )/( 4( y - C )). The second equation yields A = -B/( 2x ). Equating these two gives y = (B/2)x + C. This verifies the result obtained through linear regression. It can now be seen that the resulting family of parabolas can be classified as having vertexes which follow the graph of y = (B/2)x + C. You may ask your students if the graph could be used as a predictor of the gravitational acceleration ( or mass ) of a planet.
Finally, it might be worth pointing out to your students that the above approach to mathematics, which uses reiterations of one basic equation, has become a powerful tool for the analysis of real world events within the past few decades. This approach has become much more accessible with the introduction of computers and graphing calculators into those disciplines which seek to explore and explain the behavior of the myriads of complex systems which we find in the world around us.