Immanent Squares

'Immanent Squares' is one of two related phenomena which illustrate the presence of the perfect in the imperfect (the other is Immanent Triangles). This phenomenon allows the construction of squares from quadrilaterals of every shape.

Fig.1

 

  1. Draw any irregular quadrilateral.

  2. Construct a square above each side and find the centre point of each square (using diagonals) > EFGH.

  3. Connect the four points to a quadrilateral.

  4. Bisect each of the lines of the quadrilateral EFGH: The four new points KLMN form the corner points of a square = larger immanent square.

 

 

Fig.2 

 

  1. shows the same procedure, but the four squares are folded in reather than out. The result is again a square, but a smaller and differently positioned one.

Fig.3 

 

shows another method to obtain the same result. Here the sides of the quadrilateral are immediately bisected and the resulting figure (which is always at least of the regularity of a parallelogram) is opened into four squares. The centre points of these squares then produce the same square as in Fig.1 = KLMN. This method is even simpler but lacks the beauty and elegance of the first. It also misses out on the 'perfect cross' (Fig.9a and 9b).

 

Fig.4 

 

The same as in Fig.3 but folded in. The result is again the same as in Fig.2e.

Fig.5 

 

Construction of immanent squares from a re-entrant quadrilateral.

Fig.6 Construction of immanent squares from a crossed over quadrilateral.

Fig.7

 

 Construction of immanent squares from a triangular quadrilateral. 

(Point C & D coincide).

 

Fig.8 

 

 

Construction of immanent squares from a quadrilateral in one line and point. C & D coincide.

 

Fig.9

 

 

  1. Shows the construction of the 'larger immanent cross' by connecting the opposite centre points of the squares folded out.

  2. Shows the construction of the 'smaller immanent cross' by connecting the opposite centre points of the squares folded in.

Fig.10 

 

The unfolding is taken further:

 

  1. Shows the initial irregular quadrilateral, the quadrilateral EFGH (=the result of the first unfolding) and the immanent square on its bisected lines.

  2. Quadrilateral EFGH is once more folded out with squares. The centre of the squares are again made into a new quadrilateral JKLM.

  3. The lines of the same quadrilateral JKLM are bisected again and a second square appears, circumscribing the first.

  4. The process is further continued circumscribing the second arises etc.

Fig.11

 

 

  1. Demonstrates that the same process cannot be followed by further folding in: if the result of the first folding in, the quadrilateral EFGH is once more folded.

squares03.jpg