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The skeleton of a regular octahedron can be represented as a graph according to Steinitz's theorem, provided the graph is planar—its edges of a graph are connected to every vertex without crossing other edges—and 3-connected graph—its edges remain connected whenever two of more three vertices of a graph are removed.
The elements of the set correspond to the vertices, edges, faces and so on of the polytope: vertices have rank 0, edges rank 1, etc. with the partially ordered ranking corresponding to the dimensionality of the geometric elements. The empty set, required by set theory, has a rank of −1 and is sometimes said to correspond to the null polytope.
1.4 km – diameter of Dactyl, the first confirmed asteroid moon; 4.8 km – diameter of 5535 Annefrank, an inner belt asteroid; 5 km – diameter of 3753 Cruithne; 5 km – length of PSR B1257+12; 8 km – diameter of Themisto, one of Jupiter's moons; 8 km – diameter of the Vela Pulsar; 8.6 km – diameter of Callirrhoe, also known as ...
The Great Pyramid of Giza. The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and ...
The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with √ 2.5 edges of inscribed regular 5-cells [d] they form 400 irregular great hexagons. [ p ] The 120-cell also contains a compound of several of these great circle polygons in the same central plane, illustrated separately.
Set square shaped as 45° - 45° - 90° triangle The side lengths of a 45° - 45° - 90° triangle 45° - 45° - 90° right triangle of hypotenuse length 1.. In plane geometry, dividing a square along its diagonal results in two isosceles right triangles, each with one right angle (90°, π / 2 radians) and two other congruent angles each measuring half of a right angle (45°, or ...
In the descent above, we must rule out both the case y 1 = y 2 = y 3 = y 4 = m/2 (which would give r = m and no descent), and also the case y 1 = y 2 = y 3 = y 4 = 0 (which would give r = 0 rather than strictly positive). For both of those cases, one can check that mp = x 1 2 + x 2 2 + x 3 2 + x 4 2 would be a multiple of m 2, contradicting the ...
[4] For example, if a TNO is incorrectly assumed to have a mass of 3.59 × 10 20 kg based on a radius of 350 km with a density of 2 g/cm 3 but is later discovered to have a radius of only 175 km with a density of 0.5 g/cm 3 , its true mass would be only 1.12 × 10 19 kg.