Spirographs
What even is a Spirograph…
• a kids toy using hypotrochoids and
epitrochoids to make cool-looking geometric
patterns
Hypotrochoids v. Epitrochoids
• A hypotrochoid is a roulette traced by a point
attached to a circle of radius r rolling around
the inside of a fixed circle of radius R, where
the point is a distance d from the center of the
interior circle.
ellipse
• The ellipse is a special case of
the hypotrochoid when R = 2r.
Planetary orbits
• The orbits of the planets are ellipses with the Sun at
one focus, though all except Mercury (eccentric!) are
very nearly circular.
• Additionally, a planetary orbit is a flat plane; if that
orbit is inclined to the overall plane of the solar system,
in one complete circuit around the Sun the planet
moves once above and once below the solar system
plane.
• The orbits of the planets are all more or less in the
same plane -- called the ecliptic and defined by the
plane of the Earth's orbit. The ecliptic is inclined only 7
degrees from the plane of the Sun's equator.
Epitrochoid
…a roulette traced by a point attached to a
circle of radius r rolling around the outside of a
fixed circle of radius R, where the point is a
distance d from the center of the exterior circle.
– epicycloid
– The orbits of planets in the once popular
geocentric Ptolemaic system are epitrochoids
Ptolemaic system!
• Despite the fact that the Ptolemaic system
is considered geocentric, the planets'
motion was not thought to be actually
centered on the Earth. Instead, the
deferent was centered on a point halfway
between the Earth and another point called
the equant. The epicycle rotated and
revolved along the deferent with uniform
motion. The rate at which the planet
moved on the epicycle was fixed such that
the angle between the center of the
epicycle and the planet was the same as
the angle between the earth and the sun.
Spirograph Nebula – IC 418
• IC 418, also known
as Spirograph
Nebula, is
a planetary
nebula in the Milky
Way Galaxy.
• The name derives
from the intricate
pattern of the
nebula, which
resembles a pattern
which can be
created using
a Spirograph!
“Take the orbits of any two planets and draw a line
between the two planet positions every few
days. Because the inner planet orbits faster than the
outer planet, interesting patterns evolve.”
Star orbits
• If a star moves outward in the Galaxy, it feels
the gravitational force from a larger fraction of
the total mass; when it moves closer to the
center, less of the mass is exerting a force on
the object. As a result, orbits of stars are not
closed ellipses like those of the planets, but
instead more closely resemble the patterns
produced by a spirograph.
NASA’s Fermi
Gamma-Ray Space Telescope
• Gamma Rays are the category of the highest frequency, highest energy
light, invisible to human eyes, but emitted by nature’s most energetic
events, like the spinning disks around collapsed stars like neutron stars
and black holes.
• Pulsars are neutron stars, the crushed cores of massive suns that
destroyed themselves when they ran out of fuel, collapsed and exploded.
The blast simultaneously shattered the star and compressed its core into a
body as small as a city yet more massive than the sun. The result is an
object of incredible density, where a spoonful of matter weighs as much as
a mountain on Earth. Equally incredible is a pulsar's rapid spin, with typical
rotation periods ranging from once every few seconds up to hundreds of
times a second. Fermi sees gamma rays from more than a hundred pulsars
scattered across the sky.
Vela Pulsar from the Fermi Space
Telescope
• One pulsar shines especially bright for Fermi. Called Vela, it spins 11 times
a second and is the brightest persistent source of gamma rays the LAT
sees.
• The pictures on next slide renders Vela's position in a fisheye perspective,
where the middle of the pattern corresponds to the central and most
sensitive portion of the LAT's field of view. The edge of the pattern is 90
degrees away from the center and well beyond what scientists regard as
the effective limit of the LAT's vision.
• The pulsar traces out a loopy, hypnotic pattern reminiscent of art
produced by a Spirograph.
• The pattern created in the Vela movie reflects numerous motions of the
spacecraft. The first is Fermi's 95-minute orbit around Earth, but there's
another, subtler motion related to it. The orbit itself also rotates, a
phenomenon called precession. Similar to the wobble of an unsteady top,
Fermi's orbital plane makes a slow circuit around Earth every 54 days.
• http://www.youtube.com/watch?v=_QpMeEdmZPM
d
My Spirograph -- epitrochoid
Fixed circle
(radius R or A)
Moving circle
(radius r or B)
“pen” or
planet or star
My spirograph
“The orbit itself also
rotates, a phenomenon
called precession.” -NASA
What happens if I change the fixed
circle radius…
A LOT
Radius: 83
Radius: 13
What happens if I change the moving
circle radius…
A LOT
Radius: 115
Radius: 15
The orbit of a moon.
Bibliography (working)
• http://www.mathplayground.com/Spiromath.html
• http://www.cliffsnotes.com/study_guide/Structure-ofthe-Galaxy.topicArticleId-23583,articleId-23567.html
• http://malagabay.wordpress.com/2013/03/02/fermi-astudy-in-spirograph/
• http://www.nasa.gov/mission_pages/GLAST/news/spir
ograph.html
• http://en.wikipedia.org/wiki/Spirograph_Nebula
• http://wordsmith.org/anu/java/spirograph.html
• http://mrmikey.tumblr.com/post/13911405084/i-usedto-enjoy-spirograph-as-a-youngin