SUMMARY / RELATED TOPICS

120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol. It is called a C120, hyperdodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid; the boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge; the Davis 120-cell, introduced by Davis, is a compact 4-dimensional hyperbolic manifold obtained by identifying opposite faces of the 120-cell, whose universal cover gives the regular honeycomb of 4-dimensional hyperbolic space. There are 120 cells, 720 pentagonal faces, 1200 edges, 600 vertices. There are 4 dodecahedra, 6 pentagons, 4 edges meeting at every vertex. There are 3 pentagons meeting every edge; the dual polytope of the 120-cell is the 600-cell. The compound formed from the 120-cell and its dual is the compound of 600-cell.

The vertex figure of the 120-cell is a tetrahedron. The dihedral angle of the 120-cell is 144 °; the rows and columns correspond to vertices, edges and cells. The diagonal numbers say; the nondiagonal numbers say how many of the column's element occur at the row's element. Here is the configuration expanded with k-face elements and k-figures; the diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal. The 600 vertices of a 120-cell with an edge length of 2/φ2 = 3−√5 and a center-to-vertex radius of √8 = 2 √2 include all permutations of: and all permutations of where φ is the golden ratio, 1 + √5/2. Considering the adjacency matrix of the vertices representing its polyhedral graph, the graph diameter is 15, connecting each vertex to its coordinate-negation, at a Euclidean distance of 4√2 away, there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, 1 at distance 15.

The adjacency matrix has 27 distinct eigenvalues ranging from 2−3φ, with a multiplicity of 4, to 4, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, the rank of the adjacency matrix is 582; the 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient. One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, a structure of intertwining rings; the cell locations lend themselves to a hyperspherical description. Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians radiate out in 3 dimensions, converging at the fifth "south pole" cell; this skeleton accounts for 50 of the 120 cells. Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below.

With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, with the meridians passing through the center of each pentagonal face; the cells labeled "interstitial" in the following table do not fall on meridian great circles. The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell; the cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell; the 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration. Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring, they have no helical torsion.

They are all equivalent. The spiraling is a result of the 3-sphere curvature; the inner ring and the five outer rings now form 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive

Marquam Building

This article is about the demolished Marquam Building at the corner of SW 6th Avenue and Morrison Street in Portland, not the Marquam Building at 2501 SW 1st Avenue. The demolished building was replaced by the American Bank Building; the Marquam Building was an eight-story, Romanesque Revival office building in Portland, United States. Named for Philip Augustus Marquam, the building has been called Portland's first skyscraper and first modern office building; the building resembled a structure designed by Seattle architect John Parkinson and Pennsylvania architect John B. Hamme as an entry in the Portland Chamber of Commerce design competition of 1890. Philip Augustus Marquam acquired the lot at the corner of SW Sixth and Morrison from William W. Chapman in 1854 as payment of $500 in legal fees. Marquam resided on the property and constructed other dwellings, but in the late 1880s he began planning the Marquam Grand Opera House and the Marquam Building, adjoining structures that would cost him $600,000.

The Marquam Grand Opera House, a five-story structure adjoining the Marquam Building, opened in 1890 and was demolished in 1976. An early manager was future Portland mayor George Luis Baker; the opera house known under a series of names including Loews Theater, the Hippodrome, the Pantages, the Orpheum, opened to complementary reviews. A Portland newspaper, The Oregonian, called it "one of the neatest theaters of the west." Another review offered higher praise: "The Marquam...will eclipse all other such buildings in the northwest. It yields the palm to only one on the Pacific coast, the grand opera house in San Francisco, that only to a small degree as regards size." But critics were not as complementary. Opening in 1892, the Marquam Building was Portland's first modern office building; the Oregonian described the architecture as "very imposing." Another critic described it as "rather gloomy and cheerless, like so many of the office structures designed under the spell of the Richardsonian Romanesque...

It has no doubt all sorts of faults."Rather than pay high prices to local brick suppliers, Marquam started his own brickyard, he shipped cheaper bricks to Portland from San Francisco. Marquam's ownership of the building ended in foreclosure in 1908; the Marquam Building was sold in 1912 to real estate speculator Henry Pittock and publisher of The Oregonian. Pittock and his son-in-law, Frederick Leadbetter, intended to remodel the building to serve as headquarters for the newly organized Northwestern National Bank Company. Pittock hired general contractor Ernest Boyd MacNaughton to supervise the work. Part of the building collapsed during renovation because of substandard masonry used in the original construction. After the collapse, discussion focused upon the need for a newer, modern building. In a letter to the editor of The Architect and Engineer, one writer stated that "...as Portland advanced from a sleepy overgrown village to a half-grown city, the building became a home for quack doctors and patent medicine fakers..." and that the bricks used in construction were soft and of poor material.

He implied. Pittock fired MacNaughton and hired architect A. E. Doyle to demolish the Marquam Building and erect what would become the American Bank Building. History of Portland, Oregon Architecture of Portland, Oregon Architectural drawings of the Marquam Grand Opera House

The Gang That Sold America

The Gang That Sold America is a 1979 Italian "poliziottesco"-comedy film directed by Bruno Corbucci. It is the fourth chapter in the Nico Giraldi film series starred by Tomas Milian; the Italian progressive rock band Goblin created the soundtrack for the film. Tomas Milian: Nico Giraldi Enzo Cannavale: Salvatore Esposito Asha Puthli: Fiona Strike Leo Gavero: Don Vito Isa Danieli: Salvatore Esposito's wife Margherita Fumero: Maria Sole Giarra Gianni Musy: Gitto Cardone Tomas Milian, Jr.: Antonio, Salvatore Esposito's child Andrea Aureli: Don Mimì Irwin Keyes: support killer The Gang that Sold America was released on March 1, 1979 where it was distributed by Titanus. It grossed a total of 569 million Italian lire. List of Italian films of 1979 The Gang That Sold America on IMDb