
Features of a human face are shapes of the visible world which according to
concepts of classical Greek philosophy, and in particular according to Plato's
and other Greek philosophers philosophical concepts, is constructed according to
laws of geometry. Namely according to Plato's philosophical concepts the
structural building of the world is formed with ideal geometrical figures or
equipotent correct polygons, as these figures possess ideal or otherwise to tell
harmonious geometrical proportions.
Hence, features of a human face possess shapes which seem beautiful according to
geometrical proportions of correct geometrical figures, as objects of the world
seem beautiful if have correct geometrical ratios. Or otherwise it is possible
to tell that faces of people seem beautiful and harmonious if visible
proportions are based on ratios of correct geometrical figures, that it is
possible to consider in view of physiognomy, and also possible to consider in
cosmetic medicine and plastic surgery.
Figure of a heptagon.

To compare features of a human face to geometrical
proportions of a heptagon, it is necessary to correlate a heptagon with
the physiognomic scale of measurements. Namely any side of a heptagon
should be correlated with the line in top of a head, and the opposite
angle with the line of a chin, as shown at the left on the top chart.
Namely to top of a head there corresponds numerical value 0 or otherwise
to tell the initial scale counting, and to the line of a chin there
corresponds numerical value 180 or the final scale counting, and
accordingly it is necessary to compare any one side of a heptagon to the
initial counting, and the opposite angle with the final counting of the
physiognomic scale of measurements. And also it is necessary to compare any side of a heptagon to the chin final line, and the opposite angle with the initial line in top of a head, as shown at the left on the lower chart. Such comparison allows to compare shapes of a human face to geometry of a heptagon in a context of physiognomy, and also in a context of cosmetology and plastic surgery. If a distance from top of a head up to the line of a mouth is 144 Zn then facial features correspond proportional ratios of a heptagon, as 144 Zn specify the angle in point IV. Angles are designated in the Roman numerals. If a distance from top of a head up to the line of eyebrows is 64 Zn then proportions of a face too correspond proportional ratios of a heptagon, as in this case the line of eyebrows corresponds with the angle in point III. If a distance from top of a head up to the line of a nose is approximately 116 Zn then proportions of a human face too correspond harmonic ratios of a heptagon, as the line of a nose corresponds with the angle in a point VI. Namely in these cases it is possible to speak, that the human face possesses proportions which are caused by geometrical harmony of a heptagon. 
Besides if a tangent line to the most expressed point of in tip of a nose and the least expressed point in top of a nose bridge is parallel to lines IIV, VVI, IIIII, IIIV then human facial features correspond proportional ratios of a heptagon, that is in detail shown on the following chart.
The chart shows the Goethe's facial profile which is
compared to the physiognomic scale of measurements according to which
facial spheres have sizes: consciousness  68 zens; individuality  55 zens; feelings
 57 zens. And also the figure of heptagon is compared to the
physiognomic scale of measurements according to which it is possible to
see that sizes of facial spheres and levels of this face have no
concurrences to heptagonal geometrical proportions. But however the
distance from top of a head up to the line of a mouth has size of 145,8
Zn, that approximately coincides with proportions of a heptagon if to
consider that this distance should have size of 144 Zn for correct
concurrence. Namely it is possible to tell that the half face of the poet Goethe has position of the mouth which approximately corresponds proportions of a heptagon, but as a whole sizes of physiognomic spheres and levels of this face differ from geometrical proportions which are included in the heptagonal geometrical figure. 
The tangent line KI to the most expressed point in the tip of a nose and the
least expressed point in the top of a nose bridge is not parallel to line QR,
that also specifies difference of facial proportions of the Goethe's face from
proportions of a heptagon. But line EI (red color) is parallel to line QR that
corresponds proportions of a heptagon.
Line KI is the side of triangle JKI which symbolizes individuality, and line EI
is the side of triangle JEI which symbolizes psychological interrelations of
individuality with consciousness and feelings in a context of physiognomy.
The detailed information on triangles JKI and JEI, and also the information
on the physiognomic scale of measurements look on the previous pages of this
site section.
Pay attention, if line EI is parallel to line QR then the angle of
triangle JEI in point I has size approximately 51 degrees that is approximately
equal to the angle of an inclination of lateral sides in the Egyptian pyramid of Cheops
or pharaoh Khufu, namely proportions of a heptagon are a basis of
geometrical sizes in
the Great Pyramid of Cheops in which proportions of a human body are ciphered.
About what look the detailed information on pages of other website:
www.phantomgallery.64g.ru/pyramid/pyr1en.htm.
Derivative figure of a heptagon is a diheptagon (14 angles) which too is the
correct geometrical polygon and allows to compare shapes of a human face to
harmony of geometrical figures in a context of physiognomy, and also in a
context of cosmetic medicine and plastic surgery, that is shown on following charts.


The diheptagonal geometrical figure is compared to the
physiognomic scale of measurements differently rather than the
heptagonal figure. Namely the diheptagon is compared to lines of top of
a head and a chin in one case by the opposite sides that is possible to
see at the left on the top chart, and in the second case by opposite
angles that is possible to see at the left on the bottom chart, and as a
result figures have different parities with the physiognomic scale of
measurements. If borders of facial spheres and levels in a face coincide with diheptagonal angles, and also if lines EI or KI in triangles JKI and JEI are parallel to lines VIIVIII, VIIIX, XIXII, XIXIII then proportions of a human face correspond proportional ratios of a diheptagon. And also correspond geometrical proportions of a heptagon which is a variant of a diheptagon. Heptagon and diheptagon have different parities with the physiognomic scale of measurements, namely angles of two figures differently coincide with scale countings, but however it is necessary to consider proportions of two figures as common harmonics. Line XIVIII in the diheptagon repeats line IVI in the heptagon, and other lines of two figures have different geometrical parameters, but in all cases the geometry of two figures forms common complex of proportional harmonics. It is necessary for meaning during physiognomic analyses, and also during analyses of human faces in plastic surgery and cosmetology. 
Also in physiognomy, or in cosmetic medicine and plastic surgery it is necessary to consider concurrences of facial shapes to proportions of other correct geometrical polygons.
Figure of a pentagon.
If a distance from top of a head up to the line of a
nose is 111 Zn the features of a human face correspond proportions of a
pentagon, as 111 zens specify the angle in point V. If a distance from top of a head up to the line of eyebrows is 68 Zn then facial features too correspond proportional ratios of a pentagon, as in this case the line of eyebrows coincides with the angle in point II. If lines EI or KI in triangles JKI and JEI are parallel to lines IIII or IVV then facial features correspond proportional ratios of a pentagon. 
For example, Alexander the Great had facial profile which is shown on the following chart.
In the facial profile of Alexander the Great it is
possible to see that lines EI or KI are not parallel to the side PR in
pentagon (red color), but the tangent line to the nosetip and to the
most expressed point of eyebrows coincides with the side PR. It
specifies conformity of proportions in the face of Alexander the Great
and proportional ratios of a pentagon. Namely except for lines EI or KI in triangles JKI and JEI in a context of physiognomy, and from the point of view of proportional beauty of a human face in a context of plastic surgery and cosmetology it is possible to consider the tangent line to the nosetip and to the most expressed point of eyebrows, and also it is possible to consider other facial lines. It is necessary to tell that concurrences of lines in pentagon to features of a human face, and especially concurrence with the tangent line to the nosetip and the expressed point of eyebrows is a prominent feature or canon of ancient Greek sculptures, and including prominent feature of the shown sculptural profile of Alexander the Great. As the pentagon is an ideal figure from the point of view of the Greek philosophy, and also the pentagon is a symbol of number 5 which is key in Pythagorean numerology. Number 5 also symbolizes Athena who is the central goddess of the Greek mythology, and the pentagon is a basis of architectural proportions of Parthenon which was a temple devoted to Athena. 
Derivative figure of a pentagon is a dipentagon (ten angles) which too is the correct geometrical figure which should be used for analyses of facial shapes in physiognomy, and also the plastic surgery and cosmetic medicine should compare features of human faces to proportions of a dipentagon, that is shown on following charts.

Lines of the dipentagon and pentagon should be
considered as the common complex of proportional ratios, as two figures
are geometrical variants. If lines EI or KI in triangles JKI and JEI are parallel to lines VIVII or IXXI then features of a human face correspond proportional ratios of a dipentagon. And also if the tangent line to the nosetip and to the most expressed point of eyebrows, or to the most expressed point of a forehead, is parallel to lines VIVIII or IXXII then facial features correspond dipentagonal proportional ratios. If a distance from top of a head up to the line of a mouth is 145 Zn then facial features correspond proportions of a dipentagon, as the line of a mouth corresponds with the angle VIII. If a distance from top of a head up to the line of eyebrows is 62 Zn then features of a half face correspond dipentagonal proportions, as the line of eyebrows corresponds with the angle X. If a distance from top of a head up to the line of a nose is 118 Zn then features of a face correspond proportions of a dipentagon, as the line of a nose corresponds with the angle XI. 
Besides a derivative figure of a pentagon is a tripentagon (15 angles) which too can be considered at comparison of facial proportions of human faces with lines of correct geometrical polygons in physiognomy, or in cosmetology and plastic surgery, that is shown on following charts.
Angles of the tripentagon in points XIV and XX are
identical to angles of the dipentagone in points X and XI, namely
distances of these geometrical parameters from top of a head are 62 Zn
and 118 Zn. Line XIIIXVII repeats the line IXXII in dipentagone, and other geometrical parameters of lines and angles are different. But the tripentagon in aggregate with the dipentagon and pentagon form the common complex of proportional ratios. 
Designation in the Roman numerals in the shown charts are intended only to note the specified ahgles and lines, but it is possible to apply other system of designations which consider angular points of all correct polygons which are necessary for considering during physiognomic analyses, and during analyses of human faces by plastic surgeons and cosmeticians.
The following page results the information on comparisons of facial proportions of a human half face with lines and angular points of other correct equipotent geometrical polygons.
