# Ken Caviness, PhD

"Children are naturally curious about the world around them," says Ken Caviness, professor of physics. "If you still 'need to know,' you'll feel right at home in a physics class!"

He graduated with a BS from Southern Missionary College in 1982 with triple majors in physics, mathematics, and German, then continued his education at the University of Lowell, Massachusetts, receiving a PhD in physics in 1987, with emphases in relativity and nuclear physics. His thesis topic was titled “Considerations of Acceleration Effects in Relativistic Kinematics.” He is a member of several physics and mathematics societies, as well as the Esperanto League for North America.

Professor Caviness joined the faculty of Southern Adventist University in July 1996 after teaching physics and mathematics for three years at the French-speaking Université Adventiste d'Afrique Centrale in Rwanda and six years at Southwestern Adventist University. He served as the Physics Department chair for 11 years here at Southern and is now enjoying full-time teaching again.

Professor Caviness continues his research in several areas:

• Continuation of thesis research: acceleration in special and general relativity

• Periodic involvement with Professor Hefferlin’s projects of periodic systems of
molecules

• Various projects involving programming in Mathematica, particularly involving visualizations

• Causal networks of sequential substitution systems—begun at 2009 NKS Summer School
in Pisa, Italy; ongoing research project: what aspects of SSS causal networks reflect
features of the real universe?

Apart from his research, Professor Caviness enjoys family time with his wife, Claryce, and his two children. Among his hobbies are computers and programming, canoeing, and learning languages. He currently feels "fairly comfortable" speaking French, German, and the planned language Esperanto and has studied others also. He is now trying to learn Russian.

Dr. Caviness says that the only thing that can compare with the excitement of understanding a new idea is seeing that same flash of comprehension light up a student's face.

##### Professor Caviness' Publications

Caviness, Ken. "Cumulative Area under a Cycloid versus the Area of Its Rolling Circle." *The Wolfram Demonstrations Project*. N.p., 3 June 2014. Web <http://demonstrations.wolfram.com/CumulativeAreaUnderACycloidVersusTheAreaOfItsRollingCircle/>

Caviness, Ken. "Generating the Surreal Numbers." *The Wolfram Demonstrations Project*. N.p., 3 June 2014. Web.

<http://demonstrations.wolfram.com/GeneratingTheSurrealNumbers/>

Caviness, Ken. "Cyclogons." *The Wolfram Demonstrations Project*. N.p., 8 Jan. 2014. Web.

<http://demonstrations.wolfram.com/Cyclogons/>

Caviness, Ken. "Overlapping Light Colors." *The Wolfram Demonstrations Project*. N.p., 18 June 2012. Web.

<http://demonstrations.wolfram.com/OverlappingLightColors/>

Caviness, Ken. "The Ambassadors, Interactive. Artistic Work." *The Wolfram Demonstrations Project.* N.p., 10 June 2012. Web. <http://demonstrations.wolfram.com/TheAmbassadorsInteractive/>

Caviness, Ken. "Calkin-Wilf Tree of Fractions." *The Wolfram Demonstrations Project*. N.p., 30 May 2012. Web.

<http://demonstrations.wolfram.com/CalkinWilfTreeOfFractions/>

Caviness, Ken. "Numbers as Sets." *The Wolfram Demonstrations Project*. N.p., 30 May 2012. Web.

<http://demonstrations.wolfram.com/NumbersAsSets/>

Caviness, Ken. "Electron Waves in Bohr Atom." *The Wolfram Demonstrations Project*. N.p., 15 May 2012. Web.

<http://demonstrations.wolfram.com/ElectronWavesInBohrAtom/>

Caviness, Ken. "Tree of Strings." *The Wolfram Demonstrations Project*. N.p., 15 May 2012. Web.

<http://demonstrations.wolfram.com/TreeOfStrings/>

Caviness, Kenneth E. "Religión v. Ciencia? Por qué ellas no son enemigos." La Universidad
Adventista del Plata.

Argentina, Libertador San Martín (Entre Ríos). N.p., 28 June 2011. Scientific Lecture.

Caviness, Kenneth. "Indexing Strings and Rulesets." *The Mathematica Journal* 13.5 (11 May 2011): n. pag. Web.

<http://www.mathematica-journal.com/2011/05/indexing-strings-and-rulesets/>

Caviness, K. E. "The Biblical Basis of the Science". Biblical Foundations for the Integration of Faith and Learning Conference. Mexico, Cancun. 28 Feb. 2011. Scientific Lecture

Caviness, K. E. "Maze for Free the Key Puzzle." *The Mathematica Journal* 13.1 (20 Jan. 2011): n. pag. Web.

<http://www.mathematica-journal.com/2011/01/maze-for-free-the-key-puzzle/>

Caviness, K. E. "Applying Math and 'PhysicsThink' to Puzzles." Joint Physics/Mathematics Department EigenSeminar Colloquium, Andrews University. USA, Berrien Springs MI. 23 Sep. 2010. Seminar

Caviness, K. E. "God is Light: Equation or Metaphor?" Physics/Mathematics Department EigenVespers, Andrews University. USA, Berrien Springs MI. 23 Sep. 2010. Formal Lecture

Caviness, Ken. "Center of Mass of a Polygon." *The Wolfram Demonstrations Project*. N.p., 13 Jan. 2010. Web. <http://demonstrations.wolfram.com/CenterOfMassOfAPolygon/>

Caviness, Ken. "Sine Oval and Nested Trig Functions." *The Wolfram Demonstrations Project*. N.p., 16 Dec. 2009. Web.

<http://demonstrations.wolfram.com/SineOvalAndNestedTrigFunctions/>

Caviness, Ken. "Transverse Standing Waves." *The Wolfram Demonstrations Project*. N.p., 22 Nov. 2009. Web.

<http://demonstrations.wolfram.com/TransverseStandingWaves/>

Caviness, Kenneth E., and Lewis R. Caviness. "Fractional Graphs and Flowers." *The Wolfram Demonstrations Project*. N.p., 19 Nov. 2009. Web.

<http://demonstrations.wolfram.com/FractionalGraphsAndFlowers/>

Caviness, Ken. "*tq*-System Explorer." *The Wolfram Demonstrations Project*. N.p., 19 Nov. 2009. Web.

<http://demonstrations.wolfram.com/TqSystemExplorer/>

Caviness, Ken. "Primality Formal System Explorer." *The Wolfram Demonstrations Project*. N.p., 18 Nov. 2009. Web.

<http://demonstrations.wolfram.com/PrimalityFormalSystemExplorer/>

Caviness, Ken. "*pq*-System Explorer." *The Wolfram Demonstrations Project*. N.p., 13 Nov. 2009. Web.

<http://demonstrations.wolfram.com/PqSystemExplorer/>

Caviness, Ken. "MIU Explorer." *The Wolfram Demonstrations Project*. N.p., 28 Oct. 2009. Web.

<http://demonstrations.wolfram.com/MIUExplorer/>

Caviness, Ken, and Erik S. Caviness. "Extended Bead-Sort." *The Wolfram Demonstrations Project*. N.p., 23 Oct. 2009. Web. <http://demonstrations.wolfram.com/ExtendedBeadSort/>

Caviness, Kenneth E. "Causal Networks of Sequential Substitution Systems: Views of
highly non-local systems."

Area della Ricerca CNR di Pisa. Italy, Pisa. 9 Jul. 2009. Scientific Lecture

Caviness, Kenneth E. "Causal Networks of Sequential Substitution Systems: Views of
highly non-local systems."

Area della Ricerca CNR di Pisa. Italy, Pisa. 9 Jul. 2009. Poster Session

Caviness, Ken. "Visible Divisibility Tests." *The Wolfram Demonstrations Project*. N.p., 31 May 2007. Web.

<http://demonstrations.wolfram.com/VisibleDivisibilityTests/>

Caviness, Ken. "Egyptian Multiplication." *The Wolfram Demonstrations Project*. N.p., 28 May 2007. Web.

<http://demonstrations.wolfram.com/EgyptianMultiplication/>

Geach, J., C. Walters, B. James, K. Caviness, and R. Hefferlin. "Global Molecular Identification from Graphs.

Main-Group Triatomic Molecules." *Croatica Chemica Acta *75.2 (10 Jun. 2002): 383-400. Print

James, B., K. Caviness, J. Geach, C. Walters, and R. Hefferlin. "Global Molecular Identification from Graphs.

Neutral and Ionized Main-Group Diatomic Molecules." *Journal of Chemical Information and Computer* (1 Jan. 2002): 1-7. Print

#### Abstracts and Comments

** Cumulative Area under a Cycloid versus the Area of Its Rolling Circle**

A point on a circle rolling on a straight line without slipping traces out a cycloid. It has long been known that the area under a single arch of the curve is exactly three times the area of the generating circle, but only recently has it been shown that this is a special case of a relationship that holds throughout the cycloid's formation: the ratio of the areas of the cycloidal sector and circular sector remains exactly 3.

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** Generating the Surreal Numbers**

A surreal {L | R} is recursively defined as an ordered pair of sets of surreals; the first surreal is 0≡{Ø | Ø}, with both left and right sets empty; this is written as { I } for compactness. Once any surreal has been defined, it can also be included in the left or right sets, iteratively producing further surreals. The slider selects the first few generations of surreal numbers, which can be viewed as a list, a plot on the number line, or in tree or graph form, showing which surreals were used in the construction of others.

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** Cyclogons**

A vertex of a rolling regular polygon traces out a *cyclogon*. Roll the polygon to see the alternating circular sectors and triangular "footprints"
that form the polygon. Vary the number of sides of the polygon, showing visually that
the cyclogon becomes a cycloid in the limit as the number of sides increases.

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** Overlapping Light Colors**

To human eyes, all colors of visible light can be reproduced by combinations of red, green, and blue light, so the RGB system for specifying colors has a physiological basis. Use the sliders to control the position and intensity of the three light beams to see the effects of their overlap.

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** The Ambassadors, Interactive. Artistic Work**

The Ambassadors is the most famous example of anamorphosis in painting. This Demonstration illustrates the effect by changing the viewer's viewpoint and zooming in on the distorted diagonal shape at the bottom of the picture.

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** Calkin-Wilf Tree of Fractions**

The Calkin–Wilf tree contains all positive fractions in reduced form. Click the buttons to move down to the left or right, or to move up. The fraction's index is shown in decimal and binary notation to highlight the connections between the binary expansion of the index and the fraction's position in the tree.

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** Numbers as Sets**

Every child naturally learns how to count, but few non-mathematicians know that the "whole numbers" {0, 1, 2, ...} have been rigorously defined in terms of sets: each ordinal number is defined as the ordered set of all smaller ordinals. This Demonstration visually unpacks this recursive definition for the first few ordinals, letting you select an ordinal number and specifying how many levels deep the translation should be applied.

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** Electron Waves in Bohr Atom**

Quantization of orbits is an immediate consequence of treating the electron as a standing wave in the atom, with the principal quantum number being the number of complete wavelengths that fit in one orbit. This Demonstration provides three different visualizations and buttons to select the quantum number, or (in "2D" view) a random decimal to show the destructive self-interference of the wave.

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** Tree of Strings**

This Demonstration enumerates all possible strings, unbounded in both length and size of alphabet used. Click the buttons to move down to the left or right or to move up. The string's index is shown in decimal and binary notation to highlight the connections between the binary expansion of the index, the corresponding string, and its underlying integer composition. The "string list" mode shows the related enumeration of all possible ordered string lists.

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** Indexing Strings and Rulesets**

An enumeration of strings is developed, in which all strings of finite length of symbols from any alphabet appear, with no upper bounds for string length or alphabet size. A bijective indexing function and its inverse are found for the string enumeration, allowing iteration through the set of all strings, as well as identification of arbitrary strings by the associated index. The method is then extended to sequences of strings and to sequential substitution system (SSS) rulesets, providing a well-defined, relatively dense enumeration of all possible valid SSS rulesets for strings of arbitrary length and any number of symbols used in rulesets of any length, although in this case the indexing function is not one-to-one.

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** Maze for Free the Key Puzzle**

A series of guided tours showing how to visualize and solve puzzles programmatically, creating animated visualizations, showcasing various programming tricks and algorithms, and using some good old-fashioned physics problem-solving strategies with an occasional foray into abstract mathematics. Here the "Free the Key" puzzle is solved and animated together with basically equivalent (read isomorphic) alternative representations.

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** Center of Mass of a Polygon**

The formulas for the area and the first and second moments of a simple polygon (i.e., without holes or boundary intersections, but not necessarily convex) can be elegantly derived using Green's theorem, or can be built from first principles. Both the area and the various moments of a simple polygon can be written as simple functions of the coordinates of the vertices. These formulas can then be used to identify the center of mass of the polygon:

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** Sine Oval and Nested Trig Functions**

The unit circle is the set of all points *(x, y)=(cos(t), sin(t))*, but nesting the sine function in this formula, *(x, y)=(cos(t), sin(sin(sin(...sin(t)))))*, gives the sine oval, which becomes progressively more squarish with increased levels
of nesting (controlled by the slider). Change the graph to "sin" to examine the nested
sine function alone, whose squarish behavior is not shared by the other trig functions.

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** Transverse Standing Waves**

A standing wave is produced by the superposition of waves moving to the left and right,
reflecting back and forth between two fixed points. Interference effects produce both
nodes, where the two waves cancel by destructive interference, and antinodes, where
the waves reinforce by constructive interference. The possible wavelengths for standing
waves on a string, rope, or wire of length *L*, attached at both ends, require that *L* be an integer multiple of the half-wavelength. Click the "time" play/pause button
to start or stop the animation; adjust the number of half-wavelengths on the string
with the "harmonic n" slider; select the options to display: standing wave, wave moving
right, wave moving left, wave envelope (maximum displacement of standing wave), shading,
or velocity arrows for different parts of the string. In transverse waves, the actual
motion is perpendicular to the direction of wave propagation.

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** Fractional Graphs and Flowers**

It is common knowledge that every rational number (fraction) can be written as a repeating or terminating decimal. This Demonstration shows a colorful visualization of repeating expansions of rational numbers, highlighting the cycles. Choose the denominator and base with the sliders. Each view shows all (proper) fractions at once. In "flower" mode, first find the red numerator—the numerators are arranged in order counterclockwise around the circle. The first digit of the expansion is the black circled number. Then follow the arrows to find the successive digits. In "graph" mode, disconnected cycles are more clearly separated. (Terminating decimals are shown as a cycle of repeating 0s.)

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* tq*-System Explorer

*tq*-System Explorer

The *tq*-system [1] is a formal system in which theorems can be derived or generated from
axioms or from other theorems by using a rule of inference. Use only the upper sliders
to adjust the *tq*-string to test whether it is an axiom, a theorem, or neither. (Axioms must follow
the axiom schema or pattern * _{"xt−qx"}*, where

*represents a string of hyphens.) This Demonstration is a power tool for exploring this sample formal system, but the exploration process occurs in your own mind: do not use the lower slider or buttons until you have thoroughly explored the system or you short-circuit the process!*

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** Primality Formal System Explorer**

Explore the primality formal system [1] by clicking buttons to select axioms and generate theorems. First, click blue buttons on the "axioms" page to select the axioms to use, then on the "rules of inference" page click red buttons, each of which represents the creation of a new theorem of the system by applying a rule of inference to an axiom or previous theorem. The "proof graph" page shows the steps of your proofs graphically. (This is in treating the system in mechanical (M) mode. When you start thinking about what it all means—generating proofs of the primality of prime numbers, you are in intelligent (I) mode.)

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* pq*-System Explorer

*pq*-System Explorer

The *pq*-system [1] is a formal system in which theorems can be derived or generated from
axioms or from other theorems by using a rule of inference. Use only the upper sliders
to adjust the "*pq*-string" and test whether it is an axiom or a theorem or neither. (Axioms must follow
the axiom schema or pattern _{"xp−qx−",} where * _{x}* represents a string of hyphens.) This Demonstration is a power tool for exploring
this sample formal system, but the exploration process occurs in your own mind: do
not use the lower slider or buttons until you have thoroughly explored the system
or you short-circuit the process!

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** MIU Explorer**

Explore the MIU-system [1] by clicking the red buttons, each of which represents the creation of a new theorem of the system by applying a rule of inference to a previously found theorem. The main features of the display are (a) the list of rules for your reference; (b) the current theorem (framed in blue); and (c) the options possible by applying all rules in all possible ways, given as a list of four lists of red buttons.

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** Extended Bead-Sort**

Bead-Sort is a method of ordering a set of positive integers by mimicking the natural process of objects falling to the ground, as beads on an abacus slide down vertical rods. The number of beads on each horizontal row represents one of the numbers of the set to be sorted, and it is clear that the final state will represent the sorted set. In this Demonstration the sorting process can be run or stepped through, two sample sizes selected, and new random datasets generated. The "extended" (anti-gravity) mode allows the inclusion of all integers, with "negative beads" rising while "positive beads" fall.

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** Visible Divisibility Tests**

Most people are familiar with the test for divisibility by 3: add up all the digits and check whether the result (the digit sum) is divisible by 3. Other divisibility tests are less well known, such as alternately adding and subtracting digits to test for divisibility by 11. This Demonstration implements divisibility tests for dividing by 2, 3, ..., 11. In all cases the rule is given and a step-by-step application shown, either in "Long" or "Brief" form. The radio buttons choose which test to apply and the slider selects the number to test. (Clicking the "+" sign by the slider opens another panel where numbers can be entered directly or incremented/decremented.)

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** Egyptian Multiplication**

Egyptian multiplication: Under column headings {*n, m*}, put {1, *m*} as the first row of the table, then double each row to get the next row, continuing
down as long as the numbers in the first column are less than or equal to *n*. Now strike out (here shown in gray) enough of the rows so that the remaining first-column
numbers add up to *n*. (There is only one way to do this, as you can easily see: adding the left column
from the bottom to the top, but throwing away any entries that would make the sum
too big.) Now add the numbers remaining in the second column. The answer found will
be *n×m*.

Russian peasant multiplication: Start with {*n, m*} and generate each row from the previous one by halving the first number (discarding
any remainder) while doubling the second, until the first-column entry becomes 1.
Then strike out any rows with even numbers in the first column, and add the remaining
second-column numbers.

The two algorithms are closely related, and enabling the "explain" checkbox will show
expansions of the second-column entries which may aid in understanding why the algorithms
work. Hint: try writing *n* in binary!

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** Global Molecular Identification from Graphs. Main-Group Triatomic Molecules**

It is required that molecules with a given graph and with covalent, coordinate-covalent, and ionic bonding contain closed-shell atoms. This requirement results in an equation for each atom, which states that the number of valence electrons pertaining to it before bonding, plus those made available to it in the bonding processes, close its valence shell. Solving the equations results in identifying the atoms and the bond orders of the Lewis diagrams. The algebraic procedure can identify new species. Some of them may be considered impossible (for instance, with high steric strain), or may be transitory, or may be found only under the most unusual conditions. Lists of triatomic molecules, clusters, and resonances found by solving the equations is presented. The code for the computer program that identifies the species is listed. Closed-shell molecules lie on parallel planes in their chemical spaces, namely those on which isoelectronic molecules are located.

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** Global Molecular Identification from Graphs. Neutral and Ionized Main-Group Diatomic
Molecules**

Diophantine equations and inequalities are presented for main-group closed-shell diatomic molecules. Specifying various bond types (covalent, dative, ionic, van der Waals) and multiplicities, it becomes possible to identify all possible molecules. While many of the identified species are probably unstable under normal conditions, they are interesting and present a challenge for computational or experimental analysis. Ionized molecules with net charges of -1, 1, and 2 are also identified. The analysis applies to molecules with atoms from periods 2 and 3 but can generalized by substituting isovalent atoms. When closed-shell neutral diatomics are positioned in the chemical space (with axes enumerating the numbers of valence electrons of the free atoms), it is seen that they lie on a few parallel isoelectronic series.

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