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The Convex Hull of the Compound of the Snub Cube and Its Dual, the Pentagonal Icositetrahedron

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The Convex Hull of the Compound of the Snub Cube and Its Dual, the Pentagonal Icositetrahedron

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

The Sixth Stellation of the Triakis Octahedron

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The Sixth Stellation of the Triakis Octahedron

The triakis octahedron, a Catalan solid, is the dual of the truncated cube. When stellated six times, the triakis octahedron yields this polyhedral compound with three parts. The parts themselves appear to be unusual, irregular, dipolar octahedra with eight kites for faces, each in sets of four, with their smallest angles meeting at one vertex. However, given that these vertices are, in each case, hidden under the other parts of the compound, there is uncertainty in this.

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

On the Geography of Eurasia, and Its Major Divisions

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On the Geography of Eurasia, and Its Major Divisions

By any reasonable non-political definition, Eurasia is a single continent. Its area is 54,759,000 km², which is over one-third the earth’s total land area.

The politics of history have created, however, the “continents” of Europe, with an area of 10,180,000 km² (18.59% of Eurasia), and Asia, with an area of 44,579,000 km² (81.41% of Eurasia). These figures for Asia’s land area include that of the “subcontinent,” India, which has an area of 4,400,000 km². (Note: the subcontinent of India is a geographical term, and does not match the borders of the nation of India perfectly. The major reason for this is that India the subcontinent includes the nations of Pakistan and Bangladesh, in addition to the politically-defined nation of India.)  The subcontinent’s area is 8.04 % that of Eurasia, and 9.87% that of Asia.

Europe is a large peninsula, a part of Eurasia with a sizeable portion of its area. So is the Indian subcontinent. So, for that matter, are the Southern portions of both South America and Africa, yet no one calls them separate continents, nor even subcontinents.

Giving India a special designation of “subcontinent” makes no sense, nor does the designation of Europe as a separate continent. Both are simply parts of Eurasia.

A Half-Solved Mystery: Rotating a Sine Wave

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A Half-Solved Mystery

A few minutes ago, I wondered how to write a function whose graph would be a sine curve, but one that undulated above and below the diagonal line y=x, rather than the x-axis, as is usually the case. How to accomplish such a 45 degree counterclockwise rotation?

Well, first, I abandoned degrees, set Geometer’s Sketchpad to radians, and then simply constructed plots for both y = x and y = sin(x). Next, I added them together. The result is the green curve (and equation) you see above.

This only half-solves the problem. Does it undulate above and below y=x? Yes, it does. However, if you rotate this whole thing, clockwise, one-eighth of a complete turn, so that you are looking at the green curve going along the x-axis, you’ll notice that it is not a true sine curve, but a distorted one. Why? Because it was generated by adding y-values along the original x-axis, not by a true rotation.

I’m not certain how to correct for this distortion, or otherwise solve the problem. If anyone has a suggestion, please leave it in a comment. [Note: an astute follower of this blog has now done exactly that, so I refer the reader to the comments for the rest of the story here.]

Basic Trigonometric Functions, Viewed On a Polar Coordinate System

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Basic Trigonometric Functions, Viewed On a Polar Coordinate System

The last post made me curious about other trigonometric functions’ graphs, in a polar coordinate system. They were not what I expected. Here they are.

When A Sine Wave Is a Circle

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When A Sine Wave Is a Circle

When y=sin(x) is plotted on a polar coordinate system, with everything set, consistently, to radians, the resulting graph is a circle sitting atop the origin, with unit diameter.

Spin

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Spin

M33, the Triangulum Galaxy, Adorning the Faces of a Pentagonal Icositetrahedron

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M33, the Triangulum Galaxy, Adorning the Faces of a Pentagonal Icositetrahedron

Evidence suggests that M33 is a satellite galaxy of the even better-known Andromeda Galaxy (M31), which happens to be on a collision course with our own Milky Way. In 1.5 billion years or so, Andromeda and the Milky Way will merge to form a giant elliptical galaxy already pre-named Milkomeda. At that point, the Triangulum Galaxy may become a satellite of Milkomeda (probably one of several), or be gravitationally ejected, or simply be absorbed into Milkomeda itself.

Here, it is projected on each face of the Catalan solid which is dual to the snub cube, using software you can try at http://www.software3d.com/Stella.php.

98-Faced Polyhedron Featuring Heptagons

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98-Faced Polyhedron Featuring Heptagons

There are, as faces, 24 irregular heptagons, 6 irregular octagons of one type, and 12 of another, 24 rectangles of one type, and 24 of another, and 8 equilateral triangles. This was made using Stella 4d, which you may try or buy at http://www.software3d.com/Stella.php.

42-Faced Polyhedron Featuring Heptagons

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42-Faced Polyhedron Featuring Heptagons

The blue faces are irregular heptagons, and are twenty-four in number. There are twelve of the green rhombi, and six of the red squares. This was made using Stella 4d, which you may try or buy at http://www.software3d.com/Stella.php.