In  "Geometric Analysis of the Pentagon "  we dissected the pentagon internally and found that it is composed entirely of line segments divided into Mean and Extreme Ratio.
For those who aren't interested in the boring geometric analysis, or the math,  this paper provides a descriptive overview of the pentagon.

The sides of the pentagon are of length 1.
BF = GC =BH = JE =AF = HD = DJ = IC = AG = IE =        1/Ø
FG = FH = HJ = JI = GI =                                                      1/ز
FQ = PI = FR = MJ = HR = QG = JN = PG = HM = NI =    1/س
QR = RM = MN = NP = PQ =                                               1/Ø^4 .........
etc.
If the small pentagon in the center, MNPQR,  had its diagonals drawn they too would divide themselves in Phi ratio and so the process would continue. 1/Ø^5, 1/Ø^6, 1/Ø^7, etc. Smaller and smaller pentagons would be formed, each one rotating 36 degrees and having sides Ø times smaller than the previous one.
Where does this end? or begin?  The answer is it doesn't!
The pentagon 'ratchets' downward into infinite smallness. This can only occur because division of each line segment is absolutely perfect. Not almost perfect, but totally pefect. There is never any 'round-off' error.
Notice also that each large triangle (ADE for example) is a golden triangle.
The smaller triangles (AFG for example) are also golden mean triangles.
The pentagon itself can be considered to be a rotating golden mean triangle! If you look at triangle ADE you can see how it rotates 5 times around the center of the circle (O), each  rotation being 72 degrees (5 * 72 = 360 for the full circle). So this thing is alive! I view this just as a scientist might say that even though an object appears stationary, it is still vibrating or oscillating within itself due to 'Brownian motion.' I see the pentagon as a rotating golden mean triangle.
Imagine the pentagon rotating downwards, out of the page, becoming infinitely small. Envision ir rotating upwards out of the page, becoming infinitely large. The vertices, as they rotate form a spiral.
So the growth cyle of the pentagon is like this:
...,...1/Ø^7, 1/Ø^6,1/Ø^5,1/Ø^4,1/س,1/ز, 1/ Ø,1, Ø,ز,س,Ø^4,Ø^5,Ø^6,Ø^7,....,
On to " The Phi Ratio "...

Special characters:
\/¯  ² ³  ½  ×  Ø