Geodesic polyhedron

A skeletal polyhedron (specifically, a rhombicuboctahedron) drawn by Leonardo da Vinci to illustrate a book by Luca Pacioli

Convex polyhedron made from triangles.

- Geodesic polyhedron

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Geodesic dome

The Montreal Biosphère, formerly the American Pavilion of Expo 67, by R. Buckminster Fuller, on Île Sainte-Hélène, Montreal, Quebec
Spaceship Earth at Epcot
The Climatron greenhouse at Missouri Botanical Gardens, built in 1960 and designed by Thomas C. Howard of Synergetics, Inc., inspired the domes in the science fiction movie Silent Running.
Science World in Vancouver, built for Expo 86, and inspired by Buckminster Fuller's Geodesic dome
RISE, public art designed by Wolfgang Buttress located in Belfast consists of two spheres which also utilise Buckminster Fuller's Geodesic dome
Long Island Green Dome
Buckminster Fuller's own home, undergoing restoration after deterioration

A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron.

Blender (software)

Free and open-source 3D computer graphics software toolset used for creating animated films, visual effects, art, 3D-printed models, motion graphics, interactive 3D applications, virtual reality, and, formerly, video games.

Blender version 3.1.0 (2022)
Blender version 3.2.0 (2022)
Blender 3.1.0 splash screen
Models and render in version 2.77 (2016)
A Blender Cube (Version 2.93).
Forensic facial reconstruction of a mummy by Cícero Moraes
Geometry Nodes Editor in Blender 2.92
Physics fluid simulation
Rendering of a house
The Video Sequence Editor (VSE)
Blender's user interface underwent a significant update with Blender 2.57, and again with the release of Blender 2.80.
An architectural render showing different rendering styles in Blender, including a photorealistic style using Cycles
Using the node editor to create a moldy gold material
Game engine GLSL materials
The Dutch actor Derek de Lint in a composited live-action scene from Tears of Steel that used VFX
Example of Blender Benchmark in use
Blender 3.2.0 splash screen

Blender has support for a variety of geometric primitives, including polygon meshes, Bézier curves, NURBS surfaces, metaballs, icospheres, text, and an n-gon modeling system called B-mesh.

Goldberg polyhedron

Convex polyhedron made from hexagons and pentagons.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

A Goldberg polyhedron is a dual polyhedron of a geodesic sphere.

Geodesic grid

Screenshot of PYXIS WorldView showing an ISEA geodesic grid.
Volume rendering of Geodesic grid applied in atmosphere simulation using Global Cloud Resolving Model (GCRM). The combination of grid illustration and volume rendering of vorticity (yellow tubes). Note that for the purpose of clear illustration in the image, the grid is coarser than the actual one used to generate the vorticity.
The icosahedron
A highly divided geodesic polyhedron based on the icosahedron
A highly divided Goldberg polyhedron: the dual of the above image.
High quality volume rendering of atmosphere simulation at global scale based on Geodesic grid. The colored strips indicate the simulated atmosphere vorticity strength based on GCRM model.
A variation of geodesic grid with adaptive mesh refinement which devotes higher resolution mesh at regions of interests increasing the simulation precision while keeping the memory footage at manageable size.
High quality volume rendering of ocean simulation at global scale based on Geodesic grid. The colored strip indicate the simulated ocean vorticity strength based on MPAS model.

A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron.

Conway polyhedron notation

Used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

This example chart shows how 11 new forms can be derived from the cube using 3 operations. The new polyhedra are shown as maps on the surface of the cube so the topological changes are more apparent. Vertices are marked in all forms with circles.
Truncated tetrahedron tT
Cuboctahedron aC = aO = eT
Truncated cube tC
Truncated octahedron tO = bT
Rhombicuboctahedron eC = eO
truncated cuboctahedron bC = bO
snub cube sC = sO
icosidodecahedron aD = aI
truncated dodecahedron tD
truncated icosahedron tI
rhombicosidodeca­hedron eD = eI
truncated icosidodecahedron bD = bI
snub dodecahedron sD = sI
Triakis tetrahedron kT
Rhombic dodecahedron jC = jO = oT
Triakis octahedron kO
Tetrakis hexahedron kC = mT
Deltoidal icositetrahedron oC = oO
Disdyakis dodecahedron mC = mO
Pentagonal icositetrahedron gC = gO
Rhombic triacontahedron jD = jI
Triakis icosahedron kI
nI = kD
Deltoidal hexecontahedron oD = oI
Disdyakis triacontahedron mD = mI
Pentagonal hexecontahedron gD = gI
tI = zD
Square tiling Q = dQ
Truncated square tiling tQ
Tetrakis square tiling kQ
Snub square tiling sQ
Cairo pentagonal tiling gQ
Hexagonal tiling H = dΔ
Trihexagonal tiling aH = aΔ
Truncated hexagonal tiling tH
Rhombitrihexagonal tiling eH = eΔ
Truncated trihexagonal tiling bH = bΔ
Snub trihexagonal tiling sH = sΔ
Triangle tiling Δ = dH
Rhombille tiling jΔ = jH
Triakis triangular tiling kΔ
Deltoidal trihexagonal tiling oΔ = oH
Kisrhombille tiling mΔ = mH
Floret pentagonal tiling gΔ = gH
A 1x1 regular square torus, {4,4}{{sub|1,0}}
A regular 4x4 square torus, {4,4}{{sub|4,0}}
tQ24×12 projected to torus
taQ24×12 projected to torus
actQ24×8 projected to torus
tH24×12 projected to torus
taH24×8 projected to torus
kH24×12 projected to torus

Operators in the triangular family can be used to produce the Goldberg polyhedra and geodesic polyhedra: see List of geodesic polyhedra and Goldberg polyhedra for formulas.

Goldberg–Coxeter construction

Graph operation defined on regular polyhedral graphs with degree 3 or 4.

(3,2) master triangle over triangular grid
(3,2) master square over square grid
Master square (1,1)
Starting polyhedron (Cube)
<math>G_0</math>, the skeleton of the cube
Intermediate step of constructing <math>GC_{1,1}(G_0)</math>.
The result <math>GC_{1,1}(G_0)</math>, after rearrangement
Embedding of the result (rhombic dodecahedron)

The GC construction can be thought of as subdividing the faces of a polyhedron with a lattice of triangular, square, or hexagonal polygons, possibly skewed with regards to the original face: it is an extension of concepts introduced by the Goldberg polyhedra and geodesic polyhedra.

Hexapentakis truncated icosahedron

Convex polyhedron constructed as an augmented truncated icosahedron.

Illustration of a convex set which looks somewhat like a deformed circle. The line segment, illustrated in black above, joining points x and y, lies completely within the set, illustrated in green. Since this is true for any potential locations of any two points within the above set, the set is convex.

It is geodesic polyhedron {3,5+}3,0, with pentavalent vertices separated by an edge-direct distance of 3 steps.

List of geodesic polyhedra and Goldberg polyhedra

A skeletal polyhedron (specifically, a rhombicuboctahedron) drawn by Leonardo da Vinci to illustrate a book by Luca Pacioli

This is a list of selected geodesic polyhedra and Goldberg polyhedra, two infinite classes of polyhedra.


United States Department of Energy national laboratory specializing in high-energy particle physics.

A satellite view of Fermilab. The two circular structures are the Main Injector Ring (smaller) and Tevatron (larger).
A satellite view of Fermilab. The two circular structures are the Main Injector Ring (smaller) and Tevatron (larger).
Robert Rathbun Wilson Hall
Prototypes of SRF cavities to be used in the last segment of PIP-II Linac
Muon g−2 building (white and orange) which hosts the magnet
Transportation of the 600 ton magnet to Fermilab
Interior of Wilson Hall
Two ion sources at the center with two high-voltage electronics cabinets next to them<ref name=35years>{{cite web |title=35 years of H{{sup|−}} ions at Fermilab |url= |website=Fermilab |access-date=12 August 2015 |url-status=live |archive-url= |archive-date=18 October 2015}}</ref>
Beam direction right to left: RFQ (silver), MEBT (green), first drift tube linac (blue)
A 7835 power amplifier that is used at the first stage of linac
A 12 MW klystron used at the second stage of linac
A cutaway view of the 805 MHz side-couple cavities<ref>{{cite conference |last1=May |first1=Michael P. |last2=Fritz |first2=James R. |last3=Jurgens |first3=Thomas G. |last4=Miller |first4=Harold W. |last5=Olson |first5=James |last6=Snee |first6=Daniel |title=Mechanical construction of the 805 MHz side couple cavities for the Fermilab Linac upgrade |conference=Linear Accelerator Conference |date=1990 |journal=Proceedings of the 1990 Linear Accelerator Conference |url= |access-date=13 August 2015 |location=Albuquerque, New Mexico, USA |url-status=live |archive-url= |archive-date=7 July 2015}}</ref>
Booster ring<ref>{{cite web |title=Wilson Hall & vicinity |url= |website=Fermilab |access-date=12 August 2015 |url-status=live |archive-url= |archive-date=17 September 2015}}</ref>
Fermilab's accelerator rings. The main injector is in the foreground, and the antiproton ring and Tevatron (inactive since 2011) are in the background.

One can also find structural examples of the DNA double-helix spiral and a nod to the geodesic sphere.

Humanity Star

Reflective passive satellite designed to produce visible, pulsing flares.

The satellite was shaped like a geodesic sphere about 3 ft in diameter, with its 76 reflective panels on the shape make it looks similar to a large disco ball.