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Hexafoil

Hexafoil is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hexafoil rather than just read about it. In short: The hexafoil is a design with six-fold dihedral symmetry composed from six vesica piscis lenses arranged radially around a central point, often shown enclosed in a circumference of another six lenses. It is also sometimes known as a "daisy wheel".

Hexafoil — main illustration
Hexafoil — illustration

Key takeaways

  • Hexafoil belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Hexafoil to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Hexafoil from memory before moving on to harder problems.

Reference excerpt

The hexafoil is a design with six-fold dihedral symmetry composed from six vesica piscis lenses arranged radially around a central point, often shown enclosed in a circumference of another six lenses. It is also sometimes known as a "daisy wheel". A second, quite different, design is also sometimes referred to by this name; see alternate symbol. The design is found as a rosette ornament in artwork dating back to at least the Late Bronze Age.

Construction

The pattern figure can be drawn by pen and compass, by creating seven interlinking circles of the same diameter touching the previous circle's center. The second circle is centered at any point on the first circle. All following circles are centered on the intersection of two other circles.

The design is sometimes expanded into a regular overlapping circles grid. Bartfeld (2005) describes the construction: "This design consists of circles having a 1-[inch] radius, with each point of intersection serving as a new center. The design can be expanded ad infinitum depending upon the number of times the odd-numbered points are marked off."

Usage

The hexafoil has been very widely used throughout European folk art for a very long period of time. It is attested from at least the beginning of the Late Bronze Age, represented, for example, on ornamental golden disks found in Shaft Grave III at Mycenae (16th century BC). It is also found in some Cantabrian stelae, dated to the Iron Age, as well as Norwegian bronze kettles from the same period The six-petal rosette is common in 17th to 20th century folk art throughout Europe. In Portugal, it is common to find it in medieval churches and cathedrals, as the engraved signature of a mason; but also as decoration and symbol of protection on the chimneys of old houses in Alentejo (at times together with the lauburu, or with the pentagram). In Galicia (Spain) and all the Cantabrian Mountains, hexafoils are found since the Iron Age in torc terminals and decoration, and is still used in folk art. It can also be found in the Pyrenees (Navarre, Aragon, and Catalonia). Since 2003 the hexafoil is being used as the logo of the Alt Pirineu Natural Park, the largest of Catalonia. In the United Kingdom the hexafoil is commonly found on churches, but also in barns and private buildings, as well as on cross slabs. The use of the hexafoil as a folk magic symbol was brought from the United Kingdom to Australia by settlers, where six leaf designs with concentric circles have been found in homes and occasionally in public buildings to serve as a sign of protection. The hexafoil was also widely used on gravestones in Colonial America, especially popular in parts of Connecticut, Massachusetts, and Pennsylvania. The design was commonly used from the later 17th century until the early 19th century. The design is also known as "Sun of the Alps" (Sole delle Alpi) in Italy from its widespread use in alpine folk art. It resembles a pattern often found in that area on buildings. It is used in the coat of arms of Lecco Province. It has also been used as the emblem of Padanian nationalism in northern Italy since the 1990s. In 2001, Editoriale Nord, the publishing company of La Padania, registered the green-on-white design as a trademark. In Norway it can mostly be found on wooden objects, such as beer bowls, clothes smoothing boards, milk butts, wooden chests, beds, and so on, but it can also be found on the doors of buildings. In Norwegian it's sometimes known as "Olavsrose" (rose of Olaf), although that name is used for another symbol as well. In Lithuania the hexafoil was found on wooden beer bowls, spindles, and other wooden objects. It is known as "little sun" (saulute) in Lithuanian. In Slovakia, southeastern Poland and western Ukraine, the mark was commonly carved on roof beams inside peasant huts. In Ukraine it was known as "the symbol of Perun" (Peruna znak) and "the thunder mark" (gromovoi znak). In the Russian North the hexafoil was carved near the outside roof of peasant houses to protect them against lightning. The symbol was known as the thunder sign (gromovoi znak) or the thunder wheel (gromovoe koleso), and was associated with the thunder god Perun.

Gallery

Origin The origin and meaning of the symbol are not known, but many researchers have independently suggested that it is of religious origin, and very likely served as a protective symbol. There are two main theories for its meaning and origin.

Solar symbol Peralta Labrador (1989) cites a proposal according to which the design in the La Tène (Celtic) period was a solar symbol associated with the god Taranis. Other researchers have also described it as a solar symbol, but no reasoning for this has been given. However, the Lithuanian ("little sun") and Italian ("sun of the Alps") names do suggest a solar origin.

Thunder wheel Garshol (2021) suggests that the rosette is actually a wheel with spokes, and that it originally signified the Proto-Indo-European thunder god Perkwunos, later becoming associated with his various incarnations, such as Perun, Tarḫunz, Taranis, Thor, and Jupiter. The Russian and Ukrainian names of the symbol, as well as other more involved arguments, are given as rationale.

Alternate symbol

… excerpt ends here. Continue reading the full article.

Illustrations

Hexafoil: A geometrical hexafoil
A geometrical hexafoil
Hexafoil illustration
Hexafoil illustration
Hexafoil: Iberian funerary stele of Badalona, Spain (1st century BC)
Iberian funerary stele of Badalona, Spain (1st century BC)
Hexafoil illustration

Worked examples

Example 1 — a first encounter with Hexafoil

Start with the simplest possible case. Write down what Hexafoil claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hexafoil before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hexafoil ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hexafoil

In research
Hexafoil appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Hexafoil in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Hexafoil is common in secondary-school and first-year university syllabi. It links to neighbouring topics Circles, Geometric shapes, Gothic architecture, so understanding it makes those chapters shorter.
In everyday life
Look for Hexafoil outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Hexafoil in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Hexafoil means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Hexafoil out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hexafoil in simple terms?

The hexafoil is a design with six-fold dihedral symmetry composed from six vesica piscis lenses arranged radially around a central point, often shown enclosed in a circumference of another six lenses. It is also sometimes known as a "daisy wheel".

Why does Hexafoil matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hexafoil?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hexafoil.

Tags

  • Circles
  • Geometric shapes
  • Gothic architecture
  • Magic symbols
  • Ornaments
  • Rotational symmetry

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