Plenary5
Co-operation Between Dynamic Geometry Systems and Computer Algebra Systems -
Investigating, Guessing, Checking and Proving with the computer
Eugenio Roanes-Lozano , Dept. Algebra, Facultad
Educación, Univ. Complutense de Madrid
Extended Abstract
Computer Algebra Systems (CASs), like Maple, Derive, Mathematica, Axiom,
Macsyma, Reduce, MuPad..., are specialised in exact and algebraic calculations. They use Exact
Arithmetic and can handle non-assigned variables (i.e. variables in the "mathematical" sense,
not in the usual sense in Computer Science). Many extensions like symbolic differentiation and
integration, linear and non-linear equation and polynomial systems solving, 2D and 3D plotting...
are usually included too.
II, Cinderella, Euklid, Dr. Geo, WinGeom..., are specialised in rule and compass Geometry.
The adjective dynamic comes from the fact that, once a construction is finished, the first
objects drawn (points) can be dragged and dropped with the mouse, subsequently changing the
whole construction. They usually incorporate animation and tracing too.
Unfortunately CASs and DGSs have evolved independently. Some CASs like
Maple include specific and powerful packages devoted to Euclidean Geometry, but no CAS has
incorporated Dynamic Geometry capabilities.
On the other hand, Dynamic Geometry Systems can't handle (at least from
the point of view of the user) non-assigned variables. Therefore, what can be saved from a
DGS is only live graphic (to be read by the DGS), a geometric algorithm (script or macro,
to be interpreted by the DGS) or a dead (fixed) graphic in one of the standard graphic
formats. More precisely, what is missing in the DGSs is the possibility to handle and
export parametric data about the plot: co-ordinates of points (allowing parameters as
co-ordinates), equations of objects (allowing parameters as coefficients), length of
objects (depending on parameters)...
Some DGSs (like Cabri Geometry II or Cinderella) include theorem-checking
capabilities. This theorem-checking is based in altering the initial data: they find
counterexamples if the result is false and suppose that the result is true if they find
no counterexample (i.e., they are not "proofs" from the mathematical point of view).
This lack of co-operation is more surprising in cases like the TI-92,
where both technologies are simultaneously available.
A straightforward application of this co-operation would be to treat
with the computer the whole mathematical process of discovery (or re-discovery):
Investigating - Guessing - Checking - Proving.
The talk will begin presenting an overview of the main capabilities
of CASs and DGSs. A basic introduction to Automatic Theorem Proving in Geometry
(Gröbner bases method and Wu's pseudoremainder method) will follow. The missing
co-operation between CASs and DGSs will be detailed afterwards. Finally, the (ideal)
whole mathematical process of discovery mentioned above will be presented. All steps will
be illustrated with adequate examples.
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The Fifth International
Conference on Technology
in Mathematics Teaching
