Aikido is often represented by three different shapes: Triangle, Circle, & Square.  There are a variety of opinions concerning what these shapes symbolize, and how they apply to aikido footwork and movement.  The following might be a useful framework for applying these shapes to your Aikido.  Keep in mind that most techniques rely on combinations of the shapes.  Additionally, individuals might find that they have a tendency to favor emphasizing one shape over another.  I have included examples of Shihan after each description, as I believe, although they utilized all shapes, their techniques favored a particular shape.   

Triangle – Sword, Irimi, Pointiness

                We use triangles for stabbing and slicing into the correct angle of attack.  Interestingly, a triangle attack can be on the center line (i.e. the combined vector of each side of an equilateral triangle, as in shomenuchi), or off one of the two vectors (i.e. coming in off a yokomen angle).  Triangles are advantageous when you have superior timing.  Striking first is the key to good triangle movement.  A tertiary benefit of a triangular shape is that it can deflect energy.  However, the point of the triangle is not to deflect energy, but to deliver it to maximum effect!  This quick delivery of energy, and thus instantaneous victory without lengthy technique, make triangles ideal for multiple attacker randori.

Shioda Shihan: http://www.youtube.com/watch?v=wgZCw88jx2c

Circle – Blending, Protection, Control

                We use circles for protecting, blending, and controlling.  The key to making circles work is connection between your center, and your opponent’s center.  Circular movement will not affect uke if connection is absent.  Circles are useful for keeping uke’s balance and maintaining control once connection has been established.  Where a triangle will finish the opponent with a concentrated point of energy, a circle will spiral them into the ground or project them out.  Adding a circle to the end of a triangle typically diminishes its full effect, and this can be used to prevent harm to the training partner.  Alternatively, adding a triangle onto the end of a circular movement can be devastating.  While the triangle will give you instantaneous victory during multiple attacker randori, circles can be used to convert one of your opponents temporarily into a human shield. 

Endo Shihan: http://www.youtube.com/watch?v=ylrcUJc7MIA&p=E2D52E049E2EE00B&playnext=1&index=67

Square – Stability, Internal Strength, Immobilization

                We use squares for rooting power, lifting, or dropping.  The key to a square is controlling all four directions at the same time.  A square can be pushed on in any direction and it will not budge.  Where triangles deliver energy into uke to make him/her weak, squares soak up energy, which is then used to strengthen us and root us to the ground.  Square movements take whatever force is coming in and redirects it into the vertical plane (either into the air or into the ground, or both).  Squares are most effective when they either appear suddenly or disappear suddenly.  Squares will fall apart if there is too much slack in the connection, as this will give uke time to adjust and regain mobility.  Their vertical nature makes squares ideal for kaeshiwaza (technique reversal) and pinning.  When combined with triangular entries, squares are ideal for moving through multiple attackers with very little movement. 

Tohei Shihan: http://www.youtube.com/watch?v=PfvTLP9S84o&feature=fvw

Yamaguchi Shihan: http://www.youtube.com/watch?v=KNGq93vQLdQ