Here is the figure of which Gerard Westendorp had spoken in a comment, namely *6 congruent tetrahedra* glued between themselves *around a space diagonal* of the cube. They are, alternating, of the 2 enantiomorphic variants of a tetrahedron with all faces right trigs, 2 (2, 2, 2sqrt(2))(area = 2) & 2 (2, 2sqrt(2),2sqrt(3))(area 2sqrt(2)); the 2 variants (laevo & dextro) are of 2 different colors.
Please be happy & enjoy being alive!
It's known that a cube can be made of 5 tetrahedra: *1 regular tetrahedron* at the center, plus *4 trirectangular tetrahedra* (*trt*) glued to the 4 faces of the former.
If the edge of the cube be 2, the edges of the *regular tetrahedron* (in the interior) will have length = 2.sqrt(2) & its volume will be 8/3. Now, the *trt* has the same base as a face of the reg. tetrahedron & 1/2 its altitude, so its volume is 4/3. Together:
8/3 (reg. tetrah.) + 4. 4/3 (trt's)= 24/3 = 8 (volume of the cube)
This is *Nordstrand*'s *weird quartic* surface. (Repost of June 4, 2016)
z= 0 > 25(x^2+y^2) +50xy -4=0 > y= -25x +-10.
Please be happy & enjoy being alive, everyone!
[This is respectfully dedicated to the memory of my dearest friend, Ms *Nastasya Lee*, who made me the honor to appreciate my contributions to this community. Thank you.]
4th solid of the infinite family which begins with 15, 65, 175 & now 369 soccerballs. Each solid, with the form of a *rhombic dodecahedron*, has all the former ones inscribed as Russian dolls. The formula for the number of C60 is N= a.x^3 + b.x^2 + c*x + d.
The final formulae are:
N= 4.x^3 + 6.x^2 + 4.x + 1= (2.x+1)(2.x^2 + 2.x +1) & the genus:
g= 12. x^3 + 6.x^2 = 6.x^2(2.x+1)
voila! I thank you all for reading this and please be happy you all! New subject in next post...
Another C80 [V= 80, F= 42, E= 120] different from those both that I posted here 2 weeks ago; but this one has one nice property: if 12 such are glued face-to-face such that every one is surrounded by 5 others maintaining the Ih maximal symmetry, the structure closes up on itself and forms the elegant complex you can see [V= 780, F= 444, E= 1260; genus: g= 19]; *symmetry* is *Ih* always; please be happy and enjoy this beautiful [Sunday]!..
Want a nice solid? here it is a *propello truncated icosahedron*, this one without the unhappy solitary 5-gons of the *exp-join-6propello-h12*; face vector: [V= 240, F= 212, E= 450]; symmetry *I* (*icosahedral chiral*); faces (decreasing area): 20 6-gons (brown);12 5-gons (red); 60 4-gons (dk pink)+ 60 4-gons (lt-pink)+ 60 4-gons (blue); that's all, folks!
please be happy everybody & enjoy [autumn] while cold in not too much!
[this was posted the 1st time in the Google-plus "Mathematics community" in Nov 6, 2015]
Feuerbach's theorem (bis)
The so-called *9-point circle* is, according to *Feuerbach's theorem*, tangent to the *inscribed circle* (*10th point*) & to the *3 escribed circles* of the triangle, so it should be called the 13-point circle, at the very least.. (lol) [cf.http://arxiv.org/pdf/1107.1152.pdf ]
*Feuerbach's theorem*: The middle points of the sides of a triangle, the feet of the altitudes & the points at mid-distance from the orthocenter & the vertices are on a circle whose radius is 1/2 of the one of the circumscribed circle & whose center is the point at mid-distance from the orthocenter & the center of the circumcircle. And this circle is tangent to the inscribed circle & to the 3 escribed circles of the triangle.
This is the exact same polyhedron that I posted here half an hour ago, just with a number of *stroke-lines*, which are NOT *edges* of the polyhedron shown (they are mainly (though not always) lines where meet two faces of the same color which are not in the same plane). Edges are always *full white lines*.
Again, please be happy, everyone & enjoy being alive!
The pic today is of a handsome solid by the not easy name of *rhombic diskaiheptacontahedron* (72 faces, all *rhombi* of 2 kinds, 18 narrow ones (dark brown) & 54 wide ones (light brown); the face vector of the 'animal' is [V=74, F=72, E=144] and *symmetry* is *3-fold dihedral* with *vertical reflection* - *D3d* with Rv.
Please be happy you all & enjoy being alive!
I am an ordinary man, who *lives* & *loves* passionately every moment of his life!...
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