The Sky You Can Hold

Move the sky

Choose where you are and a time. The seven points below come from the seven Hipparcos rows printed further down, without hidden replacement coordinates. Drag the disk itself, use the time control, or press an arrow key while the canvas is focused.

Manufacture three sheets

Sheets 1 and 2 carry the kit's 150 mm ruler, plus a 100.00 mm horizontal line and a 50.00 × 50.00 mm square. Print at 100% / actual size. Turn off fit, shrink and scale-to-page options. Drivers commonly rescale without making the change obvious.

1. Print and measure first. Do not cut a sheet that the verdict refuses.
2. Optionally glue sheet 1 to cereal-box card. Sheet 2 must remain transparent: use printer-safe transparency film. Cut both 170 mm circles and pierce the marked centres. Do not cut away the altitude curves.
3. Put the transparent plate over the star disk and join the centres with a pin pushed into a folded-tape pad, or use a split pin.
4. Cut the quadrant. Tape a straight drinking straw to the sight line. Tie about 25 cm of thread at the vertex and tape a coin to its free end.
5. Set the disk's numbered rim to the triangular index on sheet 2 at the printed mean local sidereal angle. That setting is valid only for the UTC instant and longitude printed on sheet 2.

First anchor: Ptolemy's numbers, left imperfect

The earliest extant description confirmed for this build is Ptolemy's Planisphere, surviving through an Arabic textual history. That does not settle who invented stereographic projection. Sidoli and Berggren's critical translation flags garbling and small discrepancies, so the table keeps the transmitted sexagesimal values beside modern evaluation instead of forcing a match.

R = 60 parts; ε = 23;51,20°; φ_Rhodes = 36°
r(δ) = R tan((90° − δ)/2)
horizon radius = R / |sin φ|; centre offset = R cot φ

Residual means modern equation minus the printed source value, in Ptolemy's parts. It is evidence of the textual and rounding seam, not observational error.

Second anchor: the seven rows that become ink

These are ESA Hipparcos Main Catalogue fields at catalogue epoch J1991.25 in ICRS. ICRS orientation is consistent with J2000; these are not therefore “J2000 positions.” Johnson V magnitude sizes only three coarse symbol classes. It does not predict tonight's perceived brightness, and some sources vary or are multiple systems.

The disk uses exactly the displayed RA and declination. It does not propagate proper motion, precession, nutation or aberration. Even fast Arcturus moves only about 0.02° from 1991.25 to 2026 by the catalogue proper-motion scale, below a 1° paper tick. Refraction is also omitted from the prediction. Near the horizon it is variable and roughly 0.5° under standard conditions, so the field test excludes altitudes below 15°.

Why the paper can compute

The sophisticated dismissal is fair: a decorative star wheel can still look persuasive. This one earns its altitude curves from the complete circle calculation.

1. Put the unit celestial sphere at the origin, north at Z = +1, and project from the south pole onto Z = 0. With printed equator radius R, a point on paper is q = R cos δ/(1 + sin δ), x = q cos α, y = q sin α.
2. The inverse is X = 2Rx/(R²+ρ²), Y = 2Ry/(R²+ρ²), Z = (R²−ρ²)/(R²+ρ²), where ρ² = x²+y².
3. Every circle on the sphere is the intersection with a plane AX+BY+CZ=D. Substitution and collection gives:

(C + D)(x² + y²) − 2R(Ax + By) − R²(C − D) = 0

When C + D is nonzero, completing the square gives an ordinary Euclidean circle. When C + D is zero, the quadratic term vanishes and the image is a straight line. So “circles map to circles” needs the line exception: generalized circles, including a line when the spherical circle passes through the projection point.

For signed latitude φ and altitude h, the plane of constant altitude becomes:

(sin φ + sin h)(x²+y²) − 2R cos φ x − R²(sin φ − sin h) = 0
centre c_h = R cos φ/(sin φ + sin h)
radius a_h = R cos h/|sin φ + sin h|

At the equator the horizon is the exact limiting line x = 0. Whenever sin φ + sin h = 0, the implementation draws the corresponding line rather than pretending an enormous circle fits the sheet. Rotation turns right ascension α into hour angle H = θ − α. The fixed altitude circles then return altitude. The theorem is the mechanism.

What the ring means

The browser computes Earth Rotation Angle directly from the IERS constants 0.7790572732640 revolutions at J2000.0 and 1.00273781191135448 revolutions per UT1 day, then adds the IERS GMST precession polynomial. UTC stands in for UT1, a below-one-second approximation here. Nutation and the equation of the equinoxes are omitted, so the implementation consistently uses mean sidereal time, not apparent sidereal time.

Sheet 2 prints the selected UTC instant, longitude, and the computed mean local sidereal angle. The paper has no calendar converter: changing the observation time requires generating that sheet again. Longitude matters one degree of hour angle per degree of longitude. Civil-zone and daylight-saving conversion is deliberately outside the artifact.