spiral
Let an agent explain spirals and their main kinds, relay equations and properties from mathematical references, describe occurrences in nature and technology, and distinguish planar spirals from helices and from loose usage.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain spirals and their main kinds, relay equations and properties from mathematical references, describe occurrences in nature and technology, and distinguish planar spirals from helices and from loose usage.
A curve that winds around a central point while moving progressively farther from or closer to it, including the Archimedean spiral with constant spacing, generalised Archimedean spirals such as the lituus and the hyperbolic spiral, the logarithmic spiral found in shells and galaxies, the spiral of Theodorus built from right triangles, Fermat and Euler spirals, trumpet spirals, and rhumb lines that spiral toward the poles on a sphere; spirals appear throughout nature, art, engineering and mathematics.
What it is for: Not applicable; a family of curves.
It can be explain kinds and equations; relay properties; describe occurrences; distinguish from helices.
Distinguishing features
Radius changing with angle
Many named families
Self-similarity in logarithmic spirals
Natural occurrence
What it looks like
A curve coiling around a centre.
Physical character
Archimedean spiral: r = a + b theta equation
logarithmic spiral: r = a e^(b theta) equation
lituus: r^2 theta = a^2 equation
How it is recognised
Curve winding around a point
Archimedean, logarithmic, hyperbolic, lituus, Theodorus, Fermat, Euler, trumpet spirals, rhumb lines
A helix winds around an axis in three dimensions; a circle does not change radius
Related models
is a kind of - in registry terms
is exemplified by - with constant spacing
is contrasted with - in three dimensions
appears in - shells as logarithmic spirals
In practice
Families and kinds
Archimedean spiral
generalised Archimedean spirals including the hyperbolic spiral and lituus
logarithmic or equiangular spiral
Fermat spiral
Euler or Cornu spiral
spiral of Theodorus
trumpet and other named spirals
rhumb lines or loxodromes on spheres
Standards and regulation
No regulation; standard mathematical definitions
Failure modes and hazards
Confusing spirals with helices
Confusing spiral kinds
Overstating golden spiral claims in nature and art
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.spiral-artifact
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What a spiral is.
Mathematics.
Definition
Definition.
Definition
Definition.
- What is a spiral, and how does it differ from a helix and a circle? definition
- Is the question about a planar spiral, a helix or loose usage? boundary
Kinds
Kinds.
Kinds
Kinds.
- What are the Archimedean, logarithmic, hyperbolic, lituus, Theodorus, Fermat, Euler and trumpet spirals, and what are rhumb lines? definition
- Which entry fits the specific spiral? action
Mathematics Equations and properties.
Mathematics.
Equations
Equations.
Equations
Equations.
- What are the polar equations of the main spirals, and how are arc length and curvature computed? action
- Which references are standard? provenance
Properties
Properties.
Properties
Properties.
- What properties such as self-similarity and constant angle characterise logarithmic spirals? provenance
- Which entry fits logarithmic spiral? action
Occur Occurrences.
Science.
Nature
Nature.
Nature
Nature.
- Where do spirals occur in shells, galaxies, hurricanes and plants, and what is overstated about golden spirals? provenance
- Which sources are cited? provenance
Technology
Technology and art.
Technology
Technology.
- How are spirals used in springs, road transitions with Euler spirals, antennas, navigation with rhumb lines and art? provenance
- Which entry fits the specific application? action
Context History and culture.
Context.
History
History.
History
History.
- How did Archimedes, Theodorus, Bernoulli and others study spirals? provenance
- Which entry fits the history of geometry? action
Culture
Culture.
Culture
Culture.
- How do spirals appear in prehistoric art, symbolism and design? provenance
- Which entry fits spiral in art? action
What the second pass must settle
- Should logarithmic spiral and rhumb line be separate primary entries?
- How should mathematical references be linked?
- The registry entry has merged aliases naming specific spirals and rhumb lines; should they be split off?