User:Tomruen/Geodestic sphere

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Translation from [1].

Geode triangulation
Geode honeycomb

In mathematics, a geodestic sphere is a solid geometry, a non-regular polyhedron convex part a sphere. This model is used in buildings whose architecture follows the shape: the geodesic domes.

Quick Overview[edit]

Geode triangulation[edit]

Most geodes are built on the principle that one starts with a icosahedron.

icosahedral

Each of the vertices of the icosahedron is common to five triangular facets, adjacent pairs, and five edges (sides of the facets) extend from each of these vertices.

Every facet of the icosahedron is an equilateral triangle, we will subdivide into smaller triangles which are then deformed (by radial projection) to be brought on the sphere circumscribed to the icosahedron. Here are three examples of geodes, each corresponding to a different subdivision:

small geodeaverage geodebig geode

In the first example, we divided the edges of the faces of the icosahedron into two segments. In the second, the edges have been divided into three. Finally, in the latter, they were divided into ten segments. This is also what this latest model built Geode of Museum of Science and Industry de la Villette.

For the location of the vertices of the original icosahedron, just find where five small triangles (instead of six) share the same top!

Geode honeycomb[edit]

It is also conceivable geodes honeycomb taking the dual polyhedron. Geodes obtained by triangulation

dual geode

In the figure above (which is the larger of the dual polyhedron of the geode the last previous example based on a division into 10 segments figure), the sphere seems paved the hex s. But careful observation to discover that among these hexagons actually hide twelve pentagons corresponding to the vertices of the original icosahedron. It is impossible to cover a sphere using only hexagons, as shown in relationship Euler between numbers of faces, edges and vertices of a polyhedron whatever.
In the figure, three of the 12 pentagons are visible, a fourth, barely visible, is located near the edge of the figure, in the direction "eleven o'clock" (as the mark of a small needle shows), and finally a fifth lies on the edge of the figure, "half past three."

Principles of geometric construction of a geode[edit]

Geodesic domes are structures based on division (partition) side of a regular polyhedron whose faces are composed of equilateral triangles.

There are only three types of regular polyhedra with such equilateral sides: the tetrahedron regular (N = 3), the octahedron regular (N= 4) and icosahedron regular ( N= 5), the notation Nused here is the number of faces (and also the number of edges) who share the same vertex.

The division faces is defined by two integer parameters a and b positive or zero.

The first parameter a must be strictly positive.

The second parameter b can be null but must not be more than a.

Once selected values N then a and b, the construction of the corresponding dome that eventually we denote "Geode Mab" (where the notation M must be replaced by III, IV or V, depending on the value of the number in Roman numerals N), takes place in six steps, which we will explain in detail with an emphasis on case N = 5 (which is the vast majority of geodes) and illustrating the following three cases:  a = 7, b = 0 \, and  a = 4, b = 4 \, and finally  a = 5, b = 3 \,


Note: The above figures correspond, using the explained above, the following notation geodes:

  • In the introduction: the geode V-3-1 and its dual (in rotation)
  • In section 1.1: the geode V-1-0 (icosahedron) and geodes V-2-0, V-3-0 and V-10-0
  • In Section 1.2: the geode V-10-0 dual

From a regular icosahedron[edit]

Step 1[edit]

Géodes V1.PNG

We construct a regular polyhedron (R) corresponding to the value of N.

Step 2[edit]

in the three drawings below, the vertex C is at the top, the top left is A and B is the top right
  • We selected one of the faces of the polyhedron (R) and one of the edges of the face (which is always an equilateral triangle). Let AB and C chosen edge opposite to the edge on the top side selected.
  • It then divides the segment AB in (a + b) segments of equal length and all points on the numbers defined as follows: Point A has the number 0, the next point No. 1, the following No. 2, etc.. and the last, that is to say the point B, n ° a + b. Let  P_0, P_1, P_2, ... P_ {A-1}, P_a, P_ {a 1}, ... P_ {a + b} \, the points obtained.
  • It then traces the segment  CP_a \, .
  • Finally, we trace all segments parallel to  CP_a \, and passing through each point  P_0, P_1, P_2, ... P_ {A-1}, P_a, P_ {a 1}, ... P_ {a + b} \, , without exceeding the limits of the face ABC.

Step 3[edit]

Géodes V3.PNG
  • The same operation is repeated by changing each edge AB and the top C in BC and A, then B and C, thus obtaining a triple array of parallel and equidistant segments forming between them an angle of 60 ° and delimiting So small equilateral triangles with some (unless the parameter b = 0) are incomplete. These are the tops of these small triangles that will be used to construct the geodesic dome, including those located just riding on one of the edges of the face ABC.
  • Of course, repeat the operations described in steps 2 and 3 for all the faces of the polyhedron ( R). Recall that the tetrahedron has four faces, the octahedron faces 8 and 20-sided icosahedron.

Step 4[edit]

In the drawing above, the projection of these peaks is represented by a kind of little black pinhead.
  • Let O be the center of the sphere (S) confined to the polyhedron ( R). By the radial projection center O, is projected onto the sphere ( S) all networks obtained or, more accurately, the tops of small equilateral as steps 2 and 3 triangles, was obtained on each side of (R).

Step 5[edit]

Géodes V5.PNG
  • To form the edges of the geodesic dome-Va-b, we must connect the various peaks obtained in the previous step: however, we must interconnect the nodes that are the projection vertices belonging to the same small equilateral triangle (see step 3).

Step 6[edit]

In the three above drawings, coloring faces only serves to "make nice"

The edges obtained in the previous step form spherical triangles, which are the radial projection of small equilateral triangles resulting from the division of the initial polyhedron faces ( R).

  • To complete the layout of the geode-V a - b, just erase the traces of all operations in steps 1 to 4: the tops of the remaining spherical triangles are peaks of the geode, these summits, connected in pairs, draw the vertices and faces of the geode-Va-b.
  • If instead of the normal geode, it wants to build the corresponding dual geode must be determined on the sphere ( S) the center of each of these spherical triangles (which are "in front" of the centers of the faces the cyst has V-a-b normal) and if the center points thus obtained correspond to the normal adjacent faces cyst, these items must be attached in pairs to form the edges the dual geode. All these edges draw polygons are the faces of the dual geode, these faces are hexagons, but twelve of them are regular pentagons whose centers are located "in front" of the twelve vertices of the polyhedron generator (R).

It is of course clear the track of normal geode when you finish tracing the edges of the dual geode.

Géodes V duales.PNG

From a regular octahedron[edit]

Regular Octahedron

If we choose as starting polyhedron (R) regular octahedron (corresponding to N set to 4).

construction described above leads to step # 6 the following results:

Géodes IV.PNG
Géodes IV duales.PNG

From a regular tetrahedron[edit]

Regular Tetrahedron

Finally, if we choose as starting polyhedron (R) regular tetrahedron (corresponding to N set to 3).

the same construction leads to step # 6 the following results:

Géodes III.PNG
Géodes III duales.PNG

Some remarks geometric[edit]

When the parameter b is zero or equal to the parameter a, the geode (normal or dual) has all the symmetry properties of the polyhedron generator, for example, for the icosahedron: 15 planes of symmetry (via two opposite edges), 10 rotations of order 3 (120 ° rotation about an axis passing through the center of one of the sides 20) and rotation order 5 6 (rotation of 72 ° about an axis passing through two opposite corners)

geodes enantiomeric

However, when the parameters a and b are different and both positive, the geode loses its symmetry planes and there are two forms of type Nab geodes, which are enantiomers (ie ie symmetrical to each other in a mirror without being superimposed) to be convinced, simply swap the letters A and B in the explanations given above in steps 2 and 3 and carefully examine the corresponding figures (in the case V-5-3) or the figure below:

  • Spherical triangles obtained in step # 5 seem to be equilateral (at least when the generator polyhedron is an icosahedron) but they are not (their angles are not all equal) and their lengths are only a few few special cases;
  • Similarly, the hexagons obtained in the construction of dual geodes seem to be regular but generally are not (although they are when a = b = 1, regardless of N!)
  • According to a conjecture issued by Joseph D. Clinton but that remains to be proven, it would be possible to slightly move the vertices of the triangulated network described in steps 2 and 3 so that the edges of the dual Nab domes are all of equal length. J. D. Clinton based his belief on the fact that we discovered the existence of such "regulated" for all combinations of a and b following domes:
a + b < 4
a = 4 et b = 0
a = 2 et b = 2
a = 5 et b = 0
et enfin<
a = 3 et b = 3
avec N quelconque (égal à 3, 4 ou 5).

If you choose  a = 1 \, and  b = 0 \, , the geode V-1-0 is identical to the normal obtained polyhedron (R) ' 'initial generator. As for the V-geode 1-0 dual, the regular polyhedron dual polyhedron of the same, this is depending on the value of N, a regular dodecahedron (if N  = 5 \, ), a cube (if  N = 4 \, ) or a regular tetrahedron (if  N = 3 \, ).

The amount  a ^ 2 + ab + b ^ 2 \, that we should be called the "density" of a geode is interesting because it is the ratio of the surface triangular faces of the polyhedron (R) on the surface of small triangles obtained in the division of the faces. It intervenes in the form which gives, according to N,a and b, the number of faces F, edges A and S vertex normal and dual geodesic domes.

The edges of a normal geodesic dome (G) form a Delaunay the set of vertices, in addition, the geodesic dome of the same dual dome (G) is an partition of sphere (S) but it does not strictly correspond to Voronoi diagram of the vertices of the dome (G), especially for low values ​​of the density  a ^ 2 + ab + b ^ 2 \, .

It is not mathematically illogical to even look at another type of geodesic domes, those that could be obtained from a division (partition) faces another regular polyhedron, the cube, this division is to cut each of the six square cube mini square faces. To build such domes "quadrangulés" it is sufficient to step 2 described above, to divide one of the two diagonals of ABCD BD face (square) of the cube has  + b \, segments of equal length, then connect the top C at  P_a \, and finally draw all segments parallel to  CP_a and passing by points  P_0, P_1, P_2, ... P_ {A-1}, P_a, P_ {a 1}, ... P_ {a + b} \, , the diagonal, without exceeding the limits of the square face ABCD, then from step 3 to make a similar construction with diagonal AC and the top B and finally reproduce the grid obtained on each of the five other faces of the cube. For these domes, the "density ratio of the area of ​​the faces of the cube to the surface of the squares obtained in the division faces, would a^2 + ab + b^2\,.

Note: the form below does not include the special case of geodes "quadrangulées"

Form[edit]

To calculate the numbers F, A and S representing the numbers of faces, edges and vertices of a geodesic dome parameters N, and a b must first calculate the numbers f and D ( which respectively represent the number of sides of the regular polyhedron generator (R) and the "density" of the Division faces of the polyhedron ) using the following preliminary two formulas:

f = \frac{4N}{6-N}\, et D = a^2 + ab + b^2\,

Can then be calculated:

  • In the case of "normal" geodesic domes:
  • dans le cas des dômes géodésiques « normaux » :
F = fD\,,

A = \frac{3fD}{2}\, et S = \frac{fD}{2} + 2\,

  • In the case of geodesic domes "duals"
F = \frac{fD}{2} + 2\,

A = \frac{3fD}{2}\, et S = fD\,

Further details:

  • Faces normal domes are all of order 3 ( are triangles ) while their tops are of two types: those of order 6 (which result 6 edges ) and those of order N. Their numbers are worth:
S_6 = \frac{f(D-1)}{2}\,

S_N = \frac{f}{2} + 2 = \frac{12}{6-N}\,

  • The tops of domes are duals of order 3 (3 edges are result ) while their faces are of two types: those of order 6 (hexagons ) and those of order N ( N-sided polygon ). Their numbers are worth:
F_6 = \frac{f(D-1)}{2}\, et

F_N = \frac{f}{2} + 2 = \frac{12}{6-N}\,

  • The right with the segment  CP_a \, is with the height on the AB side of the face ABC regular polyhedron ( R) an angle  \theta \, whose sine and tangent are respectively:
\sin \theta =\frac{1}{2}\ \frac{a-b}{\sqrt{(a^2 + ab + b^2)}} \,

\operatorname{tg}\ \theta = \frac{\sqrt{3}}{2}\ \frac{a-b}{a+b}\,

Other examples of structures geode[edit]

Balls and balloons[edit]

A geode V-1-1 dual structure has exactly the balloons football used in the official compétations: on these balloons are 12 pentagons dyed black while 20 hexagons dyed white.

Balls golf are dug small cell number, shape and position of which can improve the performance of players, including professional golf balls, frequently encountered bullets with the circular arrangement of the cell reproduces faces (hexagons and pentagons) of a geode-V dual 6-0.

Molecules and Virus[edit]

Some remarkable organic compounds such as C 60 , whose structure is similar geodes V-1-1 were baptized fullerene s in honor of R. B. Fuller and they are also sometimes called "footballènes."

Most virus are "virus icosaédraux" (in English "icosahedral viruses") or more exactly "virus icosahedral nucleocapsid" their particularity is their structure], which very close to that of a geodesic dome normal or dual ( but the radial projection of the sphere (S) ) gives them stability. They always match N = 5 and most often at low values ​​of a and b.

Among the many viruses include those of hepatitis s A, B, C and E, one of the polio of the AIDS (HIV- 1), one of the yellow fever, one of the smallpox, one of the FMD, the usual virus bronchiolitis (VRS), those of common warts and foot ( HPV HPV-3 and HPV-1) that of rubella or that of "common cold"; also include the group of eight virus called herpesviridae which all have the structure V-5-0 that can induce various human diseases chickenpox, shingles, infectious mononucleosis, cold sores, neonatal herpes and STD such that genital herpes Simple and herpes cytomegalovirus.

Some of these viruses are "twisted" and thus correspond to Class III Fuller: e.g. polyomavirus and HPV, which are the type V-2-1. We even know a virus type V-10-7 (the one that plagues the seaweed Phaeocystis pouchetii).

See also[edit]