Polyhedral Projections Improve the Accurately of Mapping the Earth on a 2D Surface

Representing the Earth’s 3-dimensional surface on a 2-dimensional map is a problem that has vexed cartographers through the ages. The difficulties in creating a 2D map of the world include maintaining continental shapes, distances, areas, and relative positions so the 2D map is useful for its intended purpose.

World map circa 1630. Source: World Maps Online

In this article, we’ll look at selected classical projection schemes for creating a 2D world map followed by polyhedral projection schemes, the newest of which, the AuthaGraph World Map, may yield the most accurate maps of the world.

1. Classical Projections

To get an appreciation for the large number of classical projection schemes that have been developed to create 2D world maps, I suggest that you start with a visit to the Radical Cartography website at the following link, where you’ll find descriptions of 31 classical projections (and 2 polyhedral projections).

Now let’s take a look at the following classical projection schemes.

• 1569 Mercator projection
• 1855 Gail equal-area projection & 1967 Gail-Peters equal-area projection
• 1805 Mollweide equal-area projection
• 1923 Goode homolosine projection

Mercator projection

The Mercator world map is a cylindrical projection that is created as shown in the following diagram.

Source:https://tripinbrooklyn.files.wordpress.com/2008/04/new_world60_small.gif?w=450

Important characteristics of a Mercator map are:

• It represents a line of constant course (rhumb line) as a straight line segments with a constant angle to the meridians on the map. Therefore, Mercator maps became the standard map projection for nautical purposes.
• The linear scale of a Mercator map increases with latitude. This means that geographical objects further from the equator appear disproportionately larger than objects near the equator. You can see this in the relative size comparison of Greenland and Africa, below.

The size distortion on Mercator maps has led to significant criticism of this projection, primarily because it conveys a distorted perception of the overall geometry of the planet.

Gail equal-area projection & Gail-Peters equal-area projection

James Gail developed a cylindrical “equal area” projection that attempted to rectify the significant area distortions in Mercator projections. There are several similar cylindrical “equal-area” projection schemes that differ mainly in the scaling factor (standard parallel) used.

In 1967, German filmmaker Arno Peters “re-invented” the century old Gail equal-area projection and claimed that it better represented the interests of the many small nations in the equatorial region that were marginalized (at least in terms of area) in the Mercator projection.   Arno’s focus was on the social stigma of this marginalization. UNESCO favors the Gail-Peters projection.

Source: By Strebe – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16115242

Mollweide equal-area projection

The key strength of this projection is in the accuracy of land areas, but with compromises in angle and shape. The central meridian is perpendicular to the equator and one-half the length of the equator.  The whole earth is depicted in a proportional 2:1 ellipse

This projection is popular in maps depicting global distributions. Astronomers also use the Mollweide equal-area projection for maps of the night sky.

Source: Wikimedia Commons

An interrupted Mollweide map addresses the issue of shape distortion, while preserving the relative accuracy of land areas.

Source: http://www.progonos.com/furuti/MapProj/Normal/ProjInt/ProjIntC/projIntC.html

Goode homolosine projection

This projection is a combination of sinusoidal (to mid latitudes) and Mollweide at higher latitudes. It has no distortion along the equator or the vertical meridians in the middle latitudes. It was developed as a teaching replacement for Mercator maps. It is used by the U.S. Geologic Service (USGS) and also is found in many school atlases. The version shown below includes extensions repeating a few portions in order to show Greenland and eastern Russia uninterrupted.

Source: http://www.progonos.com/furuti/MapProj/Normal/ProjInt/ProjIntC/projIntC.html

2. Polyhedral Projections

In his 1525 book, Underweysung der Messung (Painter’s Manual), German printmaker Abrecht Durer presented the earliest known examples of how a sphere could be represented by a polyhedron that could be unfolded to lie flat for printing. The polyhedral shapes he described included the icosahedron and the cuboctahedron.

While Durer did not apply these ideas at the time to cartography, his work laid the foundation for the use of complex polyhedral shapes to create 2D maps of the globe. Several examples are shown in the following diagram.

Source: J.J. van Wijk, “Unfolding the Earth: Myriahedral Projections”

Now we’ll take a look at the following polyhedral maps:

• 1909 Bernard J. S. Cahill’s butterfly map
• 1943 & 1954 R. Buckminster Fuller’s Dymaxion globe & map
• 1975 Cahill-Keyes World Map
• 1996 Waterman polyhedron projections
• 2008 Jarke J. van Wijk myriahedral projection
• 2016 AuthaGraph World Map

Bernard J. S. Cahill’s Butterfly Map

Cahill was the inventor of the “butterfly map,” which is comprised of eight symmetrical triangular lobes. The basic geometry of Cahill’s process for unfolding a globe into eight symmetrical octants and producing a butterfly map is shown in the following diagram made by Cahill in his original 1909 article on this mapping process.

The octants were arrayed four above and four below the equator. As shown below, the octant starting point in longitude (meridian) was strategically selected so all continents would be uninterrupted on the 2D map surface. This type of projection offered a 2D world map with much better fidelity to the globe than a Mercator projection.

Cahill’s 1909 map. Source: genekeys.com

You can read Cahill’s original 1909 article in the Scottish Geographical Magazine at the following link:

http://www.genekeyes.com/CAHILL-1909/Cahill-1909.html

R. Buckminster Fuller’s Dymaxion Globe & Map

In the 1940s, R. Buckminster Fuller developed his approach for mapping the spherical globe onto a polyhedron. He first used a 14-sided cuboctahedron (8 triangular faces and 6 square faces), with each edge of the polyhedron representing a partial great circle on the globe. For each polyhedral face, Fuller developed his own projection of the corresponding surface of the globe. Fuller first published this map in Life magazine on 1 March 1943 along with cut-outs and instructions for assembling a polygonal globe.

Fuller’s 1943 Dymaxion map. Source: Life magazine

Fuller’s 1943 cuboctahedron Dymaxion globe.  Source: Life magazine

You can see the complete Life magazine article, “R. Buckminster Fuller’s Dymaxion World,” at the following link:

A later, improved version, known as the Airocean World Map, was published in 1954. This version of Fuller’s Dymaxion map, shown below, was based on a regular icosahedron, which has 20 triangular faces with each edge representing a partial great circle on a globe.

Source: http://www.genekeyes.com/FULLER/1972-BF-BNS-.25-.95.1-Sc-1.jpg

You can see in the diagram below that there are relatively modest variations between the icosahedron’s 20 surfaces and the surface of a sphere.

Source: https://sciencevspseudoscience.files.wordpress.com/2013/09/embedded_icosahedron.png

Fuller’s icosahedron Dymaxion globe.   Source: http://workingknowledge.com/blog/wp-content/uploads/2012/03/DymaxionPic.jpg

You can watch an animation of a spherical globe transforming into an icosahedron and then unfolding into a 2D map at the following link:

Cahill-Keyes World Map

The Cahill–Keyes World Map developed in 1975 is an adaptation of the 1909 Cahill butterfly map. The Cahill-Keyes World map also is a polyhedral map comprised of eight symmetrical octants with a compromise treatment for Antarctica. Desirable characteristics include symmetry of component maps (octants) and scalability, which allows the map to continue to work well even at high resolution.

Source: http://imgur.com/GICCYmz

Waterman polyhedron projection maps

The Waterman polyhedron projection is another variation of the “butterfly” projection that is created by unfolding the globe into eight symmetric, truncated octahedrons plus a separate eight-sided piece for Antarctica.  The Atlantic-centered projection and the comparable Pacific-centered projection are shown below.

Source, two maps: watermanpolyhedron.com

http://watermanpolyhedron.com/deploy/

Here the developers make the following claims:

“Shows the equator clearly, as well as continental shapes, distances (within 10 %), areas (within 10 %) angular distortions (within 20 degrees), and relative positions, as a compromise: statistically better than all other World maps.”

Myriahedral projection maps

A myriahedron is a polyhedron with a myriad of faces. This projection was developed in 2008 by Jarke J. van Wijk and is described in detail in the article, “Unfolding the Earth: Myriahedral Projections,” in the Cartographic Journal, which you can read at the following link:

https://www.win.tue.nl/~vanwijk/myriahedral/

Examples of myriahedral projections are shown below. As you can see, there are many different ways to define a 2D map using a myriahedral projection.

Source: https://www.win.tue.nl/~vanwijk/myriahedral/geo_aligned_maps.png

AuthaGraph World Map

The latest attempt to accurately map the globe on a 2D surface is the AuthaGraph World Map, made by equally dividing a spherical surface into 96 triangles, transferring it to a tetrahedron while maintaining areas proportions and unfolding it to be a rectangle. The developers explain the basic process as follows:

“…we developed an original world map called ‘AuthaGraph World Map’ which represents all oceans, continents including Antarctica which has been neglected in many existing maps in substantially proper sizes. These fit in a rectangular frame without interruptions and overlaps. The idea of this projection method was developed through an intensive research by modeling spheres and polyhedra. And then we applied the idea to cartography as one of the most useful applications.”

The AuthaGraph World Map. Source: AuthaGraph

For detailed information on this mapping process, I suggest that you start at the AuthaGraph home page:

http://www.authagraph.com/top/?lang=en

From here, select “Details” for a comprehensive review of the mapping technology behind the AuthaGraph World Map.

Also check out the 4 November 2016 article on the AuthaGraph World Map, “This Might Be the Most Accurate Map of the World,” at the following link:

Post-World War II Prefabricated Aluminum and Steel Houses and Their Relevance Today

With the decline of military aircraft production after World War II (WWII), the U.S. aircraft industry sought other opportunities for employing their aluminum, steel and plastics fabrication experience in the post-war economy. In the 2 September 1946 issue of Aviation News magazine, there was an article entitled “Aircraft Industry Will Make Aluminum Houses for Veterans,” that reported the following:

“Two and a half dozen aircraft manufacturers are expected soon to participate in the government’s prefabricated housing program.”

“Aircraft companies will concentrate on FHA approved designs in aluminum and its combination with plywood and insulation, while other companies will build prefabs in steel and other materials. Designs will be furnished to the manufacturers.”

“Nearly all war-surplus aluminum sheet has been used up for roofing and siding in urgent building projects; practically none remains for the prefab program. Civilian Production Administration has received from FHA specification for aluminum sheet and other materials to be manufactured. ….Most aluminum sheet for prefabs will be 12 to 20 gauge – .019 – .051 inch.”

Under the government program, the prefab home manufacturers were protected financially with FHA guarantees to cover 90% of costs, including a promise by Reconstruction Finance Corporation (RFC) to purchase any homes not sold. In addition, these manufacturers were to be given preferred access to surplus wartime factories that could be converted for mass-production of homes.

The business case for the post-war aluminum and steel pre-fabricated homes was that they could be sold profitably at a price that was substantially less than conventional wood-constructed homes.

Not surprisingly, building contractors were against this program to mass-produce pre-fabricated homes in factories. Moreover, local building codes and zoning ordnances were not necessarily compatible with the planned large scale deployment of mass-produced, prefabricated homes.

Now consider the most common housing problem of today, which seems to me to be a shortage of available low-cost housing. In recent years, this has sparked the “tiny home movement,” which is a social and architectural movement promoting living simply in small homes. Seventy years after the end of WW II, it may be time to reconsider the use of mass-produced, prefabricated aluminum and steel homes to address the current shortage of low-cost housing.

Let’s take a look at several of these efficient, and sometimes stylish post-war prefabricated homes:

• Beech Aircraft’s aluminum Dymaxion house
• Consolidated Vultee’s aluminum Fleet house
• Lustron’s steel houses
• Lincoln Houses Corporation’s aluminum houses
• Alcoa’s mid-century modern aluminum Care-free houses
• UK’s AIROH aluminum houses
• UK’s Arcon steel-framed houses
• French architect Jean Prouvé’s “Demountable house”

Beech Aircraft Corporation planned to mass-produce R. Buckminster Fuller’s Dymaxion house

In 1927, R. Buckminster Fuller developed plans for the Dymaxion (acronym for “dynamic, maximum, tension”) house, which was intended to be a mass-produced metal house of novel circular design.

Early interest in applying aircraft aluminum manufacturing techniques to post WWII housing construction was expressed by Beech Aircraft Corp. In 1944, Beech established a joint project with Dymaxion Dwelling Machines, Inc. (later renamed Fuller Houses, Inc.) to manufacture a prototype, updated Dymaxion house in Wichita, Kansas. The strong aluminum riveted structure and skin was built from WWII surplus material, with the aluminum-domed roof hung from a stainless steel strut; providing 1,017 ft2 of floor space. The aluminum, stainless steel and plastic house weighted about 8,000 pounds and was designed to withstand severe weather, including tornados.

Source: Aviation News magazineSource: TournaTalk, tournatalk.wordpress.comDymaxion house floor plan. Source: Pinterest

The 1 April 1946 issue of Aviation News magazine reported:

“Beech Aircraft Corp. expected to build 200 of these houses a day soon after the start of 1947, according to Herman Wolf, president of Fuller Homes, Inc., which will market the dwelling designed by R. Buckminster Fuller……..The houses will be subcontracted to construction firms which will combine aircraft technology and auto mass production methods. Wolf and Fuller see the new dwellings, which will sell for \$6,500 erected, as the answer to the veterans housing problem. City building codes are the big imponderable in forecasting the success of this dwelling.”

Only two Dymaxion houses were built. One is now in the Henry Ford Museum in Dearborn, Michigan.

and

http://b2dymaxionhouse.blogspot.com/p/mass-production.html

Consolidated Vultee Aircraft Corporation built the Fleet House

The California aircraft manufacturer Consolidated Vultee (later Convair) considered mass-producing a pre-fabricated homes for the post-WWII housing market. Collaboration with industrial designer Henry Dreyfuss and architect Edward Larrabee in 1947 led to the design of a small two-bedroom home. With kitchen appliances, kitchen and bathroom fixtures, and heating, the house was expected to sell for \$7,000 – \$8,000, including the cost of the lot.

The Fleet House in the factory. Source: www.thefleethouse.comThe Fleet House exterior view. Source: www.thefleethouse.com

Only two prototypes were manufactured in 1947.

An article by Jeffrey Head entitled “Snatched from Oblivion,” on the Metropolis website reported:

“Comprising 28 parts, the two-bedroom, one-bath structure appears larger than its 810 square feet because 75 percent of the exterior walls are windows. The remaining interior, roof, and garage walls are constructed of “lumicomb,” a lightweight material made of a cardboard-like honeycomb core bonded between sheets of high-strength aluminum, used at the time for airplane bulkheads. The lumicomb adds to the open feeling of the house by requiring less floor space than traditional wall and roof construction.”

“Because the resulting design was so unorthodox, Reginald Fleet, president of Southern California Homes Incorporated, opted for a novel way of marketing it. Fleet resided in the prototype with his wife and daughter, leaving it open for prospective buyers to see what life was like in a modern prefabricated home.”

“New owner Sergio Santino was about to close escrow and planned to raze the house until the South Pasadena Cultural Heritage Commission informed him of its significance.”

http://www.metropolismag.com/October-2006/Snatched-from-Oblivion/

http://www.thefleethouse.com

Here it is noted:

“Historically known as the “Consolidated Vultee House”, and commonly referred to as “the Fleet House”, today it may be the only structure still remaining that was designed, built and pre-assembled entirely in an aircraft factory.”

“The Fleet House is featured in Taschen’s PREFAB HOUSES 2010.  It is referenced in numerous publications documenting the history of pre-fab housing and has been photographed by noted post WWII architectural photographer Julius Shulman.”

The Fleet House today. Source: www.thefleethouse.com

Lustron Corporation offered low-maintenance steel homes

The Lustron Corporation, formed in 1947 by Carl Strandlund, received financing from RFC to mass-produce steel pre-fabricated houses in a former Curtiss-Wright aircraft factory in Columbus, OH.

Lustron homes came in 2- and 3-bedroom models ranging in size from 713 ft2 to 1,140 ft2. All homes came standard with enamel-coated steel exterior panels, enamel-coated steel shingle roof, metal ceiling tiles and metal-paneled interior walls, metal cabinets, closets with pocket doors, and service and storage areas.

Below is a 1949 photo of the prefabricated components of a Lustron house.

Source: Pinterest

Floor plan of a 2-bedroom Lustron “Esquire” model. Source: http://instanthouse.blogspot.com

A finished Lustron house. Source: Pinterest

Original plans were to manufacture more than 10,000 homes per year. Actual production was much less, with a total of 2,498 Lustron homes manufactured between 1948 and 1950. House prices were in the \$8,500 – \$9,500 range, increasing to an average of about \$10,500 by the end of 1949. This was approaching the price of a comparable, conventional, wood-constructed house.

The Lustron Corporation declared bankruptcy in 1950. The business failed because of several factors, including production delays, poor distribution strategy, and escalating prices that reduced the price advantage of a pre-fabricated house.

About 2,000 Lustron homes still exist today. You can read more on Lustron houses at the following link:

http://instanthouse.blogspot.com/2011/08/lustron-house.html

Lincoln Houses Corp. offered 2- and 3-bedroom aluminum homes

During WWII, Lincoln Industries developed processes for making structural material at low cost for radar housings. This process led to the design of 4’ x 8’ structural panels for buildings that were manufactured using the following process:

“Lincoln plastic panels are made by alternating sheets of heavy paper, cloth, or glass cloth with glue strips. When the desired thickness is obtained, the sheets are expanded on an automatic machine to form a honeycomb pattern. This honeycomb core is thoroughly impregnated with high-strength phenolic resin and then bonded between facing sheets of aluminum alloy, and the entire panel sealed with a vapor barrier.”

This material provides both great strength and high insulating properties. The bearing capacity of a two-inch thick wall panel compared favorably with the load carrying capacity of a brick wall one foot thick. The three-inch thick roof panels were designed to withstand an eight-foot snow load.

The basic house contained two bedrooms, a bath, living room, kitchen, dining room and general utility room. Under the Veteran’s Emergency Housing Program, the Lincoln pilot plant in Marion, Virginia manufactured and sold 2-bedroom homes for about \$3,500 – \$4,000 and a 3-bedroom home for about \$4,500, including, “wiring, water piping and heating,” constructed on a concrete or similar slab. These prices did not include the price of the land or the price of kitchen appliances and a hot water heater. Construction took about two days.

Source: Aviation News magazine

The house was designed for severe weather and the materials of construction provided protection against dry rot, internal condensation and termites.

By 1946, numerous Lincoln aluminum homes had been built and were in use. However, it appears that Lincoln never made the transition to large scale production in former airplane factories.

Aluminum Company of America (Alcoa) offered mid-century modern aluminum Care-Free Homes

After WWII, aluminum manufacturers were faced with large stocks of aluminum ore and decreasing orders. Like the aircraft manufacturers, Alcoa sough alternate markets for their finished aluminum products.

A decade after the end of WWII, Alcoa offered the Care-Free Home, which was a mid-century modern aluminum ranch house designed by Charles M. Goodman. Originally, Alcoa planned to build one Care-Free home in each of the 48 states to showcase the versatility of aluminum in home construction. A total of 24 Care-Free homes were built. The house has a 1,900 ft2 living area, a full basement, and a 2-car carport.

The framing is aluminum and wooden columns are clad in aluminum. The exterior is aluminum siding with big, aluminum-framed windows and sliding doors, and an aluminum front door. The roof and fascia strip also are aluminum. The originally expected price was about \$25,000, but actual cost was almost double. In the mid-1950s, the Care-Free house couldn’t compete with the lower cost of conventional wood construction.

Source: Alcoa 1957 brochure

Source: Alcoa 1957 brochure

You can download a 1957 Alcoa sales brochure on the aluminum Care-Free Home at the following link:

http://media.oregonlive.com/hg_impact/other/Care%20Free%20Home%20compressed%20version.pdf

Post-war prefabricated aluminum and steel homes in the UK

In 1944, the UK Ministry of Works held a public display at the Tate Gallery in London of five types of prefabricated homes.

• One aluminum prefab, made from surplus aircraft materials, the AIROH (Aircraft Industries Research Organization on Housing)
• One steel-framed prefab with asbestos panels, the Arcon, which was adapted from the all-steel Portal prototype
• Two timber-framed prefab designs, the Tarran and the Uni-Seco

This popular display was held again in 1945.

In comparison to the very small number of post-war aluminum and steel prefabricated homes built in the U.S., the production in the UK was very successful.

The AIROH aluminum house

An pre-fab package for an AIROH house consisted of 2,000 components that were assembled in four sections and delivered to the intended site by truck. The fully equipped bungalow weighed about 10 tons and provided 675 ft2 of living space, including a fully equipped kitchen and bath. In 1947 an AIROH home cost £1,610 (\$6,488 @ \$4.03 USD/£ in 1947) each to produce, plus cost of the land and installation. A total of 54,500 AIROH homes were constructed.

Source: Architects’ Journal, vol. 101, 1945 Apr. 19, p. 452

Source: https://peterdewrance.files.wordpress.com

Source: https://peterdewrance.files.wordpress.com

https://peterdewrance.wordpress.com/2015/08/14/prefab-days/comment-page-1/

The Arcon steel-framed house

The steel-framed Arcon prefabricated home had two bedrooms, fully equipped kitchen and bath and included steel built-in cabinets in the kitchen, bath and bedrooms. Exterior walls and roofing were made of corrugated asbestos panels. The house was manufactured in four 7ft-6in wide sections to enable road transportation to a pre-prepared site where the house was assembled. An Arcon house cost £1,209 (\$4,872 U.S. @ \$4.03 USD/£ in 1947) each to produce, plus cost of the land and installation. A total of 38,859 Arcon homes were constructed.

Arcon Mk V at Avondale Museum of Historic Buildings, UK. Source: avoncroft.org.uk

Arcon Mk V bungalow floor plan. Source: iwmes.org.uk/prefab.html

http://www.iwmes.org.uk/prefab.html

https://en.wikipedia.org/wiki/Prefabs_in_the_United_Kingdom

Post-war prefabricated metal frame homes in France

A notable French design was Jean Prouvé’s “Demountable House,” which was developed in 1946 under a commission from the Ministry of Reconstruction and Town Planning for use as temporary bungalows for post-war housing for Lorraine, France. The metal frame load-bearing structure of the Demountable House is shown in the first photo below. Panels of various types are then attached to the frame to complete the exterior of the house and any internal room partitions.

Frame for an 8 x 8 Demountable house. Source: http://www.dwell.com/rewind/article/progressive-prefabs-jean-prouvé

To demonstrate the ease with which Prouvé’s pre-fabricated house can be assembled on site, one model was built and then taken apart every day during Art Basel Miami 2013.

http://socks-studio.com/2013/12/17/jean-prouves-experimental-prefabricated-houses/

and

http://www.patrickseguin.com/en/designers/architect-jean-prouve/available-houses-jean-prouve/

In conclusion

In the U.S., the post-war mass production of prefabricated aluminum and steel houses never materialized. Lustron was the largest manufacturer with 2,498 houses. In the UK, over 93,000 prefabricated aluminum and steel houses were built as part of the post-war building boom that delivered a total of 156,623 prefabricated houses of all types between 1945 and 1951, when the program ended.

The lack of success in the U.S. arose from several factors, including:

• High up-front cost to establish a mass-production line for prefabricated housing, even in a big, surplus wartime factory that was available to the manufacturer on good financial terms.
• Immature supply chain to support factory operations.
• Ineffective distribution and delivery infrastructure.
• Unprepared local building codes and zoning ordnances.
• Opposition from construction workers and unions that did not want to lose work to factory-produced homes.
• Manufacturing cost increases, which reduced or eliminated the price advantage of the prefabricated homes.

From these post-war lessons learned, and with the renewed interest in “tiny homes”, it seems that there should be a business case for a modern, scalable, smart factory for the low-cost mass-production of durable prefabricated houses manufactured from aluminum, steel, and/or other materials. These prefabricated houses should be modestly-sized, modern, attractive, and customizable to a degree while respecting a basic standard design. These houses should be designed for siting on small lots in urban or suburban areas and for rapid construction.

The UK post-WWII prefab housing boom lasted seven years and delivered low-cost housing for about a half million people. I believe that there is a large market in the U.S. for this type of low-price housing, but great obstacles must be overcome, especially in California, where nobody will want a modest prefabricated home sited next to their McMansion.