Huygens and Wren created diagrams to model the conservation of momentum and energy in 1D (head on) elastic collisions.  We have extended these diagrams to model 2D collision, inelastic impacts and glancing interactions.  The diagrams use geometric constraints to encode the conservation laws.  This diagram shows an amplified rebound situation.  Hold a table tennis ball (blue) above a gulf ball (red) and drop them both on to the floor (green).  How high will the table tennis ball rebound?  Surprisingly, it will be much higher than the altitude from which they are dropped.

Interactive examples

Cheng, P. C.-H. (1996). Scientific discovery with law encoding diagrams. Creativity Research Journal, 9(2&3), 145-162.
Cheng, P. C.-H. (1996). Learning qualitative relations in physics with Law Encoding Diagrams. In G. W. Cottrell (Ed.), Proceedings of the Eighteenth Annual Conference of the Cognitive Science Society (pp. 512-517). Hillsdale, NJ: Lawrence Erlbaum.
Cheng, P. C.-H., & Simon, H. A. (1995). Scientific discovery and creative reasoning with diagrams. In S. Smith, T. Ward & R. Finke (Eds.), The Creative Cognition Approach (pp. 205-228). Cambridge, MA: MIT Press.

AVOW (Amps, Volts, Ohms and Watts) diagrams were invented to model electrical circuits.  They show values of the four electrical properties and encode Ohm's law, the power law and both of Kirchhoff's laws in the geometric and topological structure of the diagram.  AVOW diagrams substantially improve problem solving and learning of electricity compared to conventional algebra based approaches.  The example shows how current and voltage are both redistributed when one lamp in a circuit burns out. 

Cheng, P. C.-H. (2002). Electrifying diagrams for learning: principles for effective representational systems. Cognitive Science, 26(6), 685-736.
Cheng, P. C.-H., & Shipstone, D. M. (2003). Supporting learning and promoting conceptual change with box and AVOW diagrams. Part 1: Representational design and instructional approaches. International Journal of Science Education, 25(2), 193-204.
Cheng, P. C.-H., & Shipstone, D. M. (2003) . Supporting learning and promoting conceptual change with box and AVOW diagrams. Part 2: Their impact on student learning at A-level. International Journal of Science Education, 25(3), 291-305.
Shipstone, D. M., & Cheng, P. C.-H. (2002). Electric circuits: A new approach - part 2. School Science Review, 83(304), 73-81.
Shipstone, D. S., & Cheng, P. C.-H. (2001). Electric circuits: A new approach - part 1. School Science Review, 83(303), 55-63.

Probability space (PS) diagrams integrate set theoretic ideas with the axioms and laws of probability theory.  PS diagrams substantially improve problem solving and learning of probability theory compared to conventional algebra based approaches.  The diagram shows the solution to the infamous Monty Hall dilemma. 

Interactive examples

Cheng, P. C.-H. (2011). Probably good diagrams for learning: Representational epistemic re-codification of probability theory Topics in Cognitive Science 3(3), 475-498. doi: 10.1111/j.1756-8765.2009.01065.x
Cheng, P. C. H. (2003). Diagrammatic re-codification of probability theory: A representational epistemological study. In Proceedings of the Twenty Fifth Annual Conference of the Cognitive Science Society. Mahwah, NJ: Lawrence Erbaum.
Cheng, P. C.-H., & Pitt, N. G. (2003). Diagrams for difficult problems in probability. Mathematical Gazette, 87(508), 86-97.

This notation system was designed to make introductory algebra more comprehensible and easier to learn.  Its key features include: just one occurrence of a letter for each type of variable in an expression; explicit representation of the hierarchical structure of expressions; distinct network patterns for different types of relations; coherent integration of imaginary numbers with real numbers.  The top diagram gives is a HANDi for a quadratic equation.  The bottom diagram shows how imaginary numbers are encoded in a HANDi and makes transparent the relation between imginary numbers and real positive and negative numbers. 

To see a working paper contact Peter Cheng.

Truth Diagrams, TDs, are models of logical states of affairs that simultaneously encode the assignment of truth-values to variables and the truth-values of operators applied to those variables.  TDs provide an algorithmic approach to derivation of formulas and proofs by the composition of its diagrams.  The diagram shows a TD proof for a particular sequent.

To see a working paper contact Peter Cheng.