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By Hadi Mousawi · October 6, 2026
The figure-eight of Sun and Earth, why planetary spheres appear to reverse, and the interactive GA 323 geometry behind the path.
Figure · The Shape of the Path
When you observe the Sun and Earth from outside the Solar system, tracking their true positions in three-dimensional space rather than relative to each other, neither body moves in a circle. The Sun curves one loop of a figure-eight while the Earth curves the other. The two loops share a single crossing point, and the entire double shape is called a lemniscate (from the Latin for ribbon). The Winter Solstice marks the moment when the paths cross: Sun and Earth at their closest geometric meeting.
This is the picture Steiner drew in his astronomy course (GA 323) and Roland Schrapp later worked out geometrically — a study form for the Sun–Earth relationship, held beside the measured sky. The planets’ calculated positions in this app come from the ephemeris, never from these drawings.
The Sun’s analemma: one year of apparent motion traced against the sky, the earthly trace of a lemniscatory advance.
Figure · The Four Festivals
Figure · Solar declination through the year
Plotted against the ecliptic, the Sun's altitude through one year traces an asymmetric lemniscate: the upper loop reaches its peak at the Summer Solstice, the lower at the Winter Solstice, and the figure crosses at the Vernal and Autumnal Equinoxes. The earthly analemma is the observational trace of this same rhythm.
Altitude from the ecliptic through one year: the Sun’s path forms a lemniscate, with the Summer Solstice at the upper extreme, the Winter Solstice at the lower, and the Vernal and Autumnal Equinoxes at the crossing.
Figure · Drawn from the Geometry
The same figure generated directly from the lemniscate curve: Sun on one lobe, Earth on the other, the single crossing at the Winter Solstice, and the whole figure held at 23.5°, the inclination of the Earth's axis against its orbital plane. That tilt is why the crossing has a season: the figure-eight and the year are the same gesture seen from two sides. Within the Sun–Earth figure, each classical planetary sphere traces its own nested figure-eight at its own scale.
Why it matters: Classical astronomy describes the Sun and Earth relationship as a simple ellipse (Copernicus and Kepler). Schrapp observed that this description is heliocentric: it locates the Sun at a fixed point and asks where the Earth goes relative to it. When you instead ask where both go relative to the galactic frame, you see the lemniscate.
Anthroposophy has long used the lemniscate as a symbol of rhythmic breathing between polarities: between the forces of the head and the limbs, between past karma and future impulse, between systole and diastole. Schrapp’s reconstruction proposes it as more than a symbol — the geometry of the Sun and Earth relationship in this picture. It remains a model: adding the Sun’s drift toward the solar apex turns any closed figure into an open helical path, and subtracting that drift recovers the familiar relative orbit.
Figure · Sun and Earth
In this picture the lemniscate does not stay in one place. Steiner asked his audience to imagine “a rotating lemniscate which at the same time moves on through space, resulting in a lemniscatory screw” (GA 323, Lecture XVII). The solar apex — the direction of the Sun’s motion relative to the nearby stars, near Hercules — gives the drift: rather than a closed loop, the path becomes a rising, helical double spiral, and the bodies never return to exactly the same point in space. Every moment of your life occurred at a unique position in the galaxy.
The double figure advances toward the Solar apex; the path never closes in one place.
Figure · The Sun's imprints
There is a point in the cosmos where the Sun and the Earth are both together, but not at the same time. When the Sun is there on its path and has left this point by a quarter of its path, the Earth begins its movement at the point that the Sun has left. After a certain time we are on the spot in cosmic space where the Sun was: we follow the Sun’s path, cross it, and are — at a certain time of the year — at the very place where the Sun has been.
Then the Sun and Earth go forward, and after a time the Earth is again practically at the spot where the Sun was. Together with the Earth, we pass in space through the spot where the Sun has been. The Sun leaves behind results of its activity in the space it has traversed, so that the Earth enters into the imprints left behind by the Sun and crosses them. Space has living spiritual content, and the Earth enters and crosses — sails through — what the Sun has called forth.
The words above are Rudolf Steiner’s: Rudolf Steiner, lecture of 1 October 1916 (GA 171).
Figure · Schrapp
Schrapp's diagram of the full path system: Earth at the crossing node, the Sun on the outer arm of the double figure, Mercury and Venus on smaller lemniscates carried within the Sun and Earth rhythm, and the classical outer planets on the enclosing loop.
Schrapp, Fig. 7: Earth at the crossing, the Sun on the outer arm, Mercury and Venus on smaller nested lemniscates, and the outer planetary spheres on the enclosing loop.
In this app
The Cosmos → Paths view renders the lemniscatory paths of all seven classical planets (Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn) as nested figure-eights. Each planet draws its own lemniscate at a different scale inside the Sun and Earth loop, according to Schrapp's stated rotation laws.
P · Perihelion. Each year there is a point where the Sun and Earth are nearest to each other. This day is now one of the first days of January, and the point where the Sun stands then on the ecliptic and zodiac lies in the constellation of Sagittarius.
A · Aphelion. Almost six months later the reverse is true and Earth and Sun are as far as possible from each other, and this Sun’s point in the zodiac then lies in Gemini.
The line of the apsides — the line from P to A — is the major axis of the ellipse on which the Earth revolves around the Sun, and within which the Sun is one of the two foci. Note that the figure represents the ellipse as much too strongly eccentric. The axes of this ellipse “rotate” in space with respect to the zodiac in a period, according to observations, of approximately 110,000 years.
Note on terms. Perihelion is the point in the orbit of a celestial object where it is closest to the Sun (helios: Greek for Sun), and perigee the point where it is closest to the Earth (geo meaning Earth). Aphelion and apogee name the farthest points in the same way.
Cosmic clock of movement of apsides against zodiac clock (precession)
The movement of the apsides shows that these points gradually change their position, just as the cardinal points of spring, summer, autumn and winter shift on account of the precession. The apsides, however, move far more slowly and in the opposite direction — that is, with the signs of the zodiac.
In the drawing, imagine the ellipse surrounded by the zodiac at an indefinite distance, and revolving with reference to the zodiac. Thus, the axes AP and KL always point to other points in the heavens, and the approach of Earth and Sun takes place under a star of the zodiac lying farther on. At the same time, the cross formed by the four cardinal points winter, spring, summer, and autumn also revolves due to the precession, against the direction of the zodiacal constellations and in the course of approx. 25,920 years. In the drawing the position is given for our time, of course rather roughly.
Another important fact is to be seen from this: with this movement toward each other of the ‘cross of the ellipse-axes’ and of the ‘cross of the year’, the single points P, A, K, L and W, Sp, S, F will meet with each other alternately. Again and again there will be a time … when the cardinal points of the year will coincide with cross of the axes, or the line of the apsides. For example the vernal equinox can coincide with the perihelion (Sp on P). The autumnal equinox and the summer and winter solstices can do the same. The opposite cardinal point will then coincide with the aphelion.
At each of the four meetings that take place approx. Every 5000 years, the Earth’s axis is always at right angles to the major axis (the apsides, coinciding with the line of the solstices) or to the minor axis (coinciding with the line of the equinoxes). … the cone that the Earth’s axis describes in 25,920 years is placed in the center of the ellipse, as are also the lines of the equinoxes and solstices. The axis of the Earth and of the equator is, however, as always at right angles to the solstice line and to the equinoctial line.
Willi Sucher, plate FMC00.613 / FMC00.613A, with Elisabeth Vreede’s astronomy letters.
Three ellipses: two of the meetings, and the present day. Hover, focus or tap an ellipse to read it.
Four interactive studies from Steiner's astronomy course: where the lemniscate comes from mathematically, how the same figure lives in the human form, how it governs eclipses, and the whole seven-planet system rendered on it.
Study A · The Curve of Cassini
Steiner introduces the lemniscate as one frame of the Curve of Cassini, the locus of a constant product of distances to two foci, just as the ellipse is the constant sum and the hyperbola the constant difference. It is not a stretched figure-eight.
Sweep the focal distance below. The oval pinches into the lemniscate (K = c), then splits into two disjoint loops (K < c), the case where, in his words, we are obliged “to go out of space… to preserve the continuity of the idea.” The dashed counter-space ghost stands in for that lost continuity.
Study B · Sky and Body
The planetary loop and the human form are one lemniscatory principle. The figure closes upward into the head, the spherical pole, where Saturn, Jupiter and Mars have their loops in opposition, and opens downward into the limbs, the radial pole of Venus and Mercury, whose loops fall in conjunction. The Sun holds the rhythmic crossing between.
Study C · Moon and Eclipses
An eclipse is the crossing of two curves: the ecliptic, the Sun–Earth plane, and the inclined path of the Moon. They meet at two nodes, and when Sun, Earth and Moon align at a node, the crossing becomes an eclipse. (Inclination shown exaggerated for legibility.)
Study D · The Full System
The whole system on this one geometry: Earth at the crossing node, a star field, the twelvefold zodiac ring, and all seven planets as nested Cassini curves. A single slider sweeps the entire system together from oval to lemniscate to split, the living metamorphosis behind the fixed loops you see in the Cosmos.
Source: the Cassini derivation and the three regimes are stated in GA 323 (L9–L10); the human-form correspondence and eclipse geometry follow L11–L12. The rendered curves are teaching-forms, not an ephemeris; the counter-space layer is an interpretive projective gesture, not a literal exit from space. See the live Cosmos for the real-time system.
Source: The lemniscate geometry and the Sun/Earth pivot table are stated by Schrapp in The Lemniscatory Path System (2012), Figures 74 to 86 (pp. 56 to 63). The advancing-path description follows directly from the stated geometry. Nested-lemniscate construction for the other planets uses Schrapp's rotation tables for each planet; continuous interpolation between his discrete monthly snapshots is this app's own reconstruction. GA 323 scenes are teaching-forms, not an ephemeris.