In this case the vertices of A 1 are the midpoints of the sides of the quadrilateral A 0 and those of A 2 are the apices of the triangles with apex angles ? / 2 erected over the sides of A 1.
12.
Kites and darts in which the two isosceles triangles forming the kite have apex angles of 2? / 5 and 4? / 5 represent one of two sets of essential tiles in the Penrose tiling, an aperiodic tiling of the plane discovered by mathematical physicist Roger Penrose.
13.
And likewise for angles " B, C ", with equality in the first part if the triangle is isosceles and the apex angle is at least 60?and equality in the second part if and only if the triangle is isosceles with apex angle no greater than 60?
14.
And likewise for angles " B, C ", with equality in the first part if the triangle is isosceles and the apex angle is at least 60?and equality in the second part if and only if the triangle is isosceles with apex angle no greater than 60?
15.
In the latter double inequality, the first part holds with equality if and only if the triangle is isosceles with an apex angle of at least 60? and the last part holds with equality if and only if the triangle is isosceles with an apex angle of at most 60?
16.
In the latter double inequality, the first part holds with equality if and only if the triangle is isosceles with an apex angle of at least 60? and the last part holds with equality if and only if the triangle is isosceles with an apex angle of at most 60?
17.
The two-stage process yielding the square A 2 can be carried out in two different ways . ( The apex " Z " of an isosceles triangle with apex angle ? erected over a line segment " XY " is the midpoint of the line segment " XY " .)
18.
One such deviation was a " double " cone, which appeared as if a cone started to grow from its tip with a certain apex angle ( e . g . 84?), but then abruptly changed the apex angle ( e . g . to 39?) in a single point on its surface, thus producing a break in the observed cross-section of the cone.
19.
One such deviation was a " double " cone, which appeared as if a cone started to grow from its tip with a certain apex angle ( e . g . 84?), but then abruptly changed the apex angle ( e . g . to 39?) in a single point on its surface, thus producing a break in the observed cross-section of the cone.
20.
:" If isosceles triangles with apex angles 2k? / n are erected on the sides of an arbitrary n-gon A 0, and if this process is repeated with the n-gon formed by the free apices of the triangles, but with a different value of k, and so on until all values 1 d " k d " n " 2 have been used ( in arbitrary order ), then a regular n-gon A n " 2 is formed whose centroid coincides with the centroid of A 0 ".
