Geometry a simple or a complex thing

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Geometry a simple or a complex thing?

• Maurice OReilly
• CASTeL, St Patricks College,
• Drumcondra

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What is Geometry?

• Hardy 1925
• Hodge 1955
• Atiyah 1982

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Hardy 1925
There is not one geometry, but an infinite
number of geometries. But I am not speaking now
to an audience of rude and simple physicists, or
of philosophers dazed by centuries of
Aristotelian tradition, but to one of
mathematicians familiar with common mathematical
ideas.
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Hardy 1925
the history of mathematics shows conclusively
that mathematicians will never accept the
tyranny of any philosopher. The moment any
philosopher has demonstrated the impossibility
of any mode of thought, some rebellious
mathematician will employ it with unconquerable
energy and conspicuous success.
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Hardy 1925
Geometries, I will ask you to agree
provisionally, are models, Attention to the
role of definitions or axioms, and theorems,
distinguishing between analytic and pure
geometries the elementary geometry of schools
is a fundamentally and inevitably illogical
subject, about whose details agreement can never
be reached.
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Hodge 1955
Gender inclusive language! Geometry had become
specialised and exhausted The tyranny of the
Erlanger Program has not yet finally
disappeared, but I am prepared to say that its
days are numbered.
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Hodge 1955
I should describe a geometry as the study of a
space with a structure. Open sets and
differentiable manifolds
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Hodge 1955
young students should understand that the
concepts with which geometry deals are part of
the whole fabric of mathematics, and that the
difference between geometry and, say, algebra is
really one of method. The equivalence of many
ways of representing a geometry deserves to be
brought out prominently.
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Atiyah 1982
Of all the changes that have taken place in the
mathematical curriculum, both in schools and
universities, nothing is more striking than the
decline in the central role of
geometry. Examples of non-physical spaces
Complex plane, Reimann surfaces, Dynamics (
rigid bodies), Line geometry, Function spaces.
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Atiyah 1982
Broadly speaking I want to suggest that geometry
is that part of mathematics in which visual
thought is dominant whereas algebra is that part
in which sequential thought is dominant. This
dichotomy is perhaps better conveyed by the words
insight versus rigour and both play an
essential role in real mathematical problems.
geometry is not so much a branch of mathematics
as a way of thinking that permeates all branches.
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Have you used?

• Geometers sketchpad
• Cabri 3D
• GeoGebra

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Texts
John Stillwell The Four Pillars of Geometry
David Henderson Diana Taimina Experiencing
Geometry Euclidean and non-Euclidean with
History
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4 Pillars

• Euclidean geometry
• Coordinate geometry/vector algebra
• Projective geometry
• Transformation geometry

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Whats happening in my GeoGebra class?

• In the lab-based session, four stages
• Gave students hard copy of diagram,
• together with instructions.
• Moved five points mentioned without revealing the
  structure behind them.
• Showed them the Construction Protocol, explaining
  the GeoGebra features new to them.
• Guided them through the construction of the
  GeoGebra worksheet, encouraging independent
  exploration.

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Static situation
Michael wrote We have a circle with centre O
and points C, B, A on the circle There is a
point E on the triangle There is a 2nd triangle
where F is a point on the triangle. Naomi wrote
(but crossed out) Mirror the triangle CBA to
the other half of the circle. Place F on mirrored
(symmetric) to the point E.
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Static situation
Ciara wrote A square inside a circle split into
two triangles with a diagonal. One triangle is
shaded red. E is a point on the shaded triangle.
F is a point on the other triangle. D is a point
on the circle. Brendan wrote 2 triangles
opposite can be seen, making up the measurements
of a circle
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Dynamic picture
Michael Move E around the triangle F also
moves on the corresponding triangle. Move the
point D, which is constrained to the circle, the
2nd triangle then moves around about the circle.
  Naomi When E is moved, F stays as the mirror
image of E, so as E moved F also moves to be
symmetric. When D moves, the mirrored triangle
also moves. Why????
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Dynamic picture
Ciara E moves so does F in the same way. D
moves so does the outlined triangle. C moves
so does one other point of the other triangle
  Brendan When E is moved, F also moves. E
moves around its triangle and so does F in its
own triangle in opposite direction
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Understanding the structure
Michael noted cMultE,D point F, when D is
moved this is the movement of F. Rotates round
centre O, I think.   Naomi cMultE,D As D
moves, F moves similar to E as E moves, F moves
in symmetry. LocusF,E GeoGebra draws the path
F takes The similar triangle copies triangle
CAB
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Understanding the structure
Ciara 2 triangles remain congruent.   Brendan
As D moves around the circle, the triangle
opposite ACB also moves. As C moves, the size
of the triangle ACB changes and the size of the
opposite triangle also changes.
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References
Learning framework in the spirit of Povey and
Angiers educative assessment. Nice article in
MSOR Connections on tools in GeoGebra by Sangwin
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Remark
GeoGebra accessible and effective software
providing an environment for students to observe,
articulate, think and do mathematics.
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Other GeoGebra explorations

• Platonic solids non-Euclidean geometry
• Projective geometry
• Desargues theorem
• Pythagoras theorem
• Golder ratio regular pentagon
• Planar graphs
• 7 circles theorem

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Plenty to do!
Go 4 it! Thank you