Figure 1: (a) Dividing a circle’s perimeter into five equal division for drawing a Pentagon;
(b) pentagon apexes and five symmetry axes
Figure 2: (a) The relationship between pentagon and pentagram; (b) Symmetry axis dm is perpendicular to the side ab at point m; (c) the harmonic diminishing pentagons
Figure 3: A method of drawing a pentagon
Figure 4: (a) Connecting each point to the next first point results in a pentagon;
(b) Connecting each point to the next second point results in a pentagram;
(c) Connecting every point to each other results in a pentagon with its pentagram
Figure 5: A method of drawing a pentagon according to Albuzjani [12]
Figure 6: The relationship between Pentagon, √5 and ϕ [13]
Figure 7: A method of drawing the pentagon in and on the triangle and square according to Albuzjani [12]
Figure 8: A star pentagon at decreasing scales of edges of the star cut each other in the Golden Section [14]
Figure 9: The five Plato solids [16,17]
Figure 10: A dodecahedron consists of twelve pentagon [15]
Figure 11: Star pentagons form a starfish [14]
Figure 12: A man as a pentagon (left); Five as the flowering or quintessence of life (right) [13]
Figure 13: Pentagonal symmetry in marine animals [17]
Figure 14: Pentagonal symmetry in flowers [17]
Figure 15: The golden divisions contained in the pentagram are shown to determine the proportion of this ancient mask of Hermes [13]
Figure 16: Pentagonal stamp tiles from small dome space of Friday mosque of Isfahan, Iran [18]
Figure 17: The pentagonal Proportion in the tomb of Imamzada Abdulla at Farsajin [19]
Figure 18: Pentagons in the shell of Gray dome in Maraqeh, Iran [20]
Figure 19: Five-fold star and ten-fold star in the wooden door of Imam Khomeini mosque in Mashahd, Iran [20]
Figure 20: Pentagon and ten-fold star in blunt knot of 2 and 5 [20]
Figure 21: Pentagon and ten-fold star in entrance gateway of Darolfonun School Tehran, Iran
Figure 22: The pentagon has been used in method of producing major knot [21]
Figure 23: (a) Dividing a circle perimeter into seven equal divisions as drawing a heptagon; (b) Heptagon has seven sides, seven apexes and axe
Figure 24: (a) Symmetry axis em is Perpendicular to the side ab at Point m; (b) the harmonic Diminishing heptagons
Figure 25: The relationship between Heptagon and seven-fold star
Figure 26: A method of drawing a heptagon
Figure 27: Notation for the diagonals, dk = 2sin πk/n of a polygon of unit Radius [19]
Figure 28: The formula to find number of diagonals of a polygon (n (n-3)/2) [19]
Figure 29: Number of parallel diagonals there are in a heptagon [14]
Figure 30: A heptagon and an octagon surround with pentagons. There is no space between pentagons around octagon
Figure 31: A circle of heptagons which created with connecting parallel diagonals
Figure 32: (a) and (b): A composition resulting from the combination of five-sided and seven-sided polygons that are covering a two-dimensional surface. In this case, the seven-sided shapes are regular but the paired five-seven shapes are not
Figure 33: (a) Another combination of five-sided and seven-sided polygons that are covering a two-dimensional surface without leaving any gap. In fact, this composition is the structure of a HC5 C7 nanotube.
(b) The difference between the irregularly produced seven-sided shape and the regular heptagon is shown