The **Waterman "Butterfly" World Map** is a map arrangement created by Steve Waterman. Waterman first published a map in this arrangement in 1996. The arrangement is an unfolding of a globe treated as a truncated octahedron, evoking the *butterfly map* principle first developed by Bernard J.S. Cahill (1866–1944) in 1909. Cahill and Waterman maps can be shown in various profiles, typically linked at the north Pacific or north Atlantic oceans.

As Cahill was an architect, his approach tended toward forms that could be demonstrated physically, such as by his flattenable rubber-ball map. Waterman, on the other hand, derived his design from his work on close-packing of spheres. This involves connecting the sphere centers from cubic closest-packed spheres into a corresponding convex hull, as demonstrated in the accompanying graphics. These illustrate the W5 sphere cluster, W5 convex hull, and two Waterman projections from the W5 convex hull.

To project the sphere to the polyhedron, the Earth is divided into eight octants. Each meridian is drawn as three straight-line segments in its respective octant, each segment defined by its endpoints on two of four "Equal Line Delineations" defined by Waterman. These Equal Line Delineations are the North Pole, the northernmost polyhedron edge, the longest line parallel to the equator, and the equator itself. The intersections of all meridians with any one Equal Line Delineation are equally spaced, and the intersections of all parallels with any one meridian are equally spaced.^{ [1] } Waterman chose the W5 Waterman polyhedron and central meridian of 20°W to minimize interrupting major land masses. Popko notes the projection can be gnomonic too.^{ [2] } The two methods yield very similar results.

Like Buckminster Fuller's 1943 Dymaxion Projection, an octahedral butterfly map can show all the continents uninterrupted if its octants are divided at a suitable meridian (in this case 20°W) and are joined, for example, at the North Atlantic, as in the 1996 version.^{ [3] }^{ [4] }

In geometry, a **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

The **Mercator projection** is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

In geometry, a **polyhedron** is a three dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as *poly-* + *-hedron*.

In cartography, a **map projection** is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. Every distinct map projection distorts in a distinct way, by definition. The study of map projections is the characterization of these distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

In geometry, the **stereographic projection** is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

A **zonohedron** is a convex polyhedron with point symmetry, every face of which is a polygon with point symmetry. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a **zonotope**.

In geometry, a **circumscribed sphere** of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word **circumsphere** is sometimes used to mean the same thing. As in the case of two-dimensional circumscribed circles, the radius of a sphere circumscribed around a polyhedron *P* is called the circumradius of *P*, and the center point of this sphere is called the circumcenter of *P*.

A **gnomonic map projection** displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the sphere passes through the point on the surface and then on to the plane. No distortion occurs at the tangent point, but distortion increases rapidly away from it. Less than half of the sphere can be projected onto a finite map. Consequently, a rectilinear photographic lens, which is based on the gnomonic principle, cannot image more than 180 degrees.

The **scale** of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

The **sinusoidal projection** is a pseudocylindrical equal-area map projection, sometimes called the **Sanson–Flamsteed** or the **Mercator equal-area projection**. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing in a world map of 1570.

**Geometric graph theory** in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices, thus it is "the theory of geometric and topological graphs".

In geometry, a **truncated 24-cell** is a uniform 4-polytope formed as the truncation of the regular 24-cell.

**Bernard Joseph Stanislaus Cahill**, American cartographer and architect, was the inventor of the octahedral "Butterfly Map". An early proponent of the San Francisco Civic Center, he also designed hotels, factories and mausoleums like the Columbarium of San Francisco.

In geometry, the **Waterman polyhedra** are a family of polyhedra discovered around 1990 by the mathematician Steve Waterman. A Waterman polyhedron is created by packing spheres according to the cubic close(st) packing (CCP), then sweeping away the spheres that are farther from the center than a defined radius, then creating the convex hull of the resulting pack of spheres.

In cartography, the **cylindrical equal-area projection** is a family of cylindrical, equal-area map projections.

The **Cahill–Keyes projection** is a polyhedral compromise map projection first proposed by Gene Keyes in 1975. The projection is a refinement of an earlier 1909 projection by Cahill. The projection was designed to achieve a number of desirable characteristics, namely symmetry of component maps (octants), scalability allowing the map to continue to work well even at high resolution, uniformity of geocells, metric-based joining edges, minimized distortion compared to a globe, and an easily understood orientation to enhance general usability and teachability.

The **octant projection** or **octants projection**, is a type of projection proposed the first time, in 1508, by Leonardo da Vinci in his Codex Atlanticus. Leonardo's authorship would be demonstrated by Christopher Tyler, who stated "For those projections dated later than 1508, his drawings should be effectively considered the original precursors..". In fact, there is a sketch of it on a page in the Codex Atlanticus manuscripts, made from the very hand of Leonardo, being Leonardo's sketch, the first known description of the *octant projection*.

**Leonardo's world map** is a map drawn in the "octant projection" in approximately 1514, found among the papers of Leonardo da Vinci. It features an early use of the name America. The map incorporates information from the travels of Amerigo Vespucci, published in 1503 and 1505. Additionally, the map accurately shows the Arctic as an ocean and Antarctica as a continent of about the correct size.

In map projections, an **interruption** is any place where the globe has been split. All map projections are interrupted at at least one point. Typical world maps are interrupted along an entire meridian. In that typical case, the interruption forms an east/west boundary, even though the globe has no boundaries.

- ↑ Steve Waterman, "Waterman Projection Method",
*Waterman Project Website* - ↑ Edward S., Popko (2012).
*Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere*. Taylor & Francis. pp. 20–21. ISBN 9781466504295. - ↑ Darvias, Gyorgy (2002).
*Symmetry: Culture and Science*. Symmetrion. pp. 129–171. ISBN 963-214-761-8. - ↑ Dongo, Studio (2013).
*The City That Traveled The World*. CreateSpace Independent Publishing Platform. pp. cover and acknowledgements page. ISBN 9781484966228.

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- Real-time winds and temperature on Waterman projection
- Interactive Waterman butterfly map
- Interactive Tissot indicatrix of Waterman projection
- Polyhedral maps
- Angular distortion of Waterman map (gnomonic projection)
- Critique of Waterman projection
- Explanation of equal-line delineation for projection

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