Hexagram

Key takeaways

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

Reference excerpt

A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. The term is used to refer to a compound figure of two equilateral triangles. The intersection is a regular hexagon. The hexagram is part of an infinite series of shapes which are compounds of two n-dimensional simplices. In three dimensions, the analogous compound is the stellated octahedron, and in four dimensions, the compound of two 5-cells is obtained. It has been historically used in various religious and cultural contexts and as decorative motifs. The symbol was used as a decorative motif in medieval Christian churches and Jewish synagogues. In the medieval period, a mystical symbol known as the Seal of Solomon was depicted as either a hexagram or a pentagram.

Group theory In mathematics, the root system for the simple Lie group G2 is in the form of a hexagram, with six long roots and six short roots.

Construction by compass and a straight edge A six-pointed star, like a regular hexagon, can be created using a compass and a straight edge:

Make a circle of any size with the compass. Without changing the radius of the compass, set its pivot on the circle's circumference, and find one of the two points where a new circle would intersect the first circle. With the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked. With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles.

Construction by linear algebra

A regular hexagram can be constructed by orthographically projecting any cube onto a plane through three vertices that are all adjacent to the same vertex. The twelve midpoints to edges of the cube form a hexagram. For example, consider the projection of the unit cube with vertices at the eight possible binary vectors in three dimensions:

( 1 , 0 , 0 ) ( 0 , 1 , 0 ) ( 0 , 0 , 1 ) ( 1 , 1 , 0 ) ( 1 , 0 , 1 ) ( 0 , 1 , 1 ) ( 1 , 1 , 1 ) {\displaystyle {\begin{array}{ccc}(1,&0,&0)\\(0,&1,&0)\\(0,&0,&1)\\(1,&1,&0)\end{array}}\qquad {\begin{array}{ccc}(1,&0,&1)\\(0,&1,&1)\\(1,&1,&1)\\&\end{array}}} onto the plane x + y + z = 1. The midpoints are

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Hexagram

Start with the simplest possible case. Write down what Hexagram claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, 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 Hexagram 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 Hexagram 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 Hexagram

In research: Hexagram appears in astronomy 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 Hexagram 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: Hexagram is common in secondary-school and first-year university syllabi. It links to neighbouring topics 6 (number), Art history, Church architecture, so understanding it makes those chapters shorter.
In everyday life: Look for Hexagram 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.

How to study Hexagram in 20 minutes

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

Frequently asked questions

What is Hexagram in simple terms?

A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. The term is used to refer to a compound figure of two equilateral triangles.

Why does Hexagram matter?

Because it connects several astronomy 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 Hexagram?

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 Hexagram.