Space diagonal

Key takeaways

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

Reference excerpt

In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the same face (but not on the same edge) as each other. For example, a pyramid has no space diagonals, while a cube (shown at right) or more generally a parallelepiped has four space diagonals.

Axial diagonal An axial diagonal is a space diagonal that passes through the center of a polyhedron. For example, in a cube with edge length a, all four space diagonals are axial diagonals, of common length a 3 . {\displaystyle a{\sqrt {3}}.} More generally, a cuboid with edge lengths a, b, and c has all four space diagonals axial, with common length a 2 + b 2 + c 2 . {\displaystyle {\sqrt {a^{2}+b^{2}+c^{2}}}.}

A regular octahedron has 3 axial diagonals, of length a 2 {\displaystyle a{\sqrt {2}}} , with edge length a. A regular icosahedron has 6 axial diagonals of length a 2 + φ {\displaystyle a{\sqrt {2+\varphi }}} , where φ {\displaystyle \varphi } is the golden ratio ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} .

Space diagonals of magic cubes

A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.

See also Distance Face diagonal Hypotenuse Magic cube classes Spacetime interval

References

John R. Hendricks, The Pan-3-Agonal Magic Cube, Journal of Recreational Mathematics 5:1:1972, pp 51–54. First published mention of pan-3-agonals Hendricks, J. R., Magic Squares to Tesseracts by Computer, 1998, 0-9684700-0-9, page 49 Heinz & Hendricks, Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0, pages 99,165 Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 173, 1994.

External links Weisstein, Eric W. "Space Diagonals". MathWorld. de Winkel Magic Encyclopedia Heinz - Basic cube parts John Hendricks Hypercubes

Illustrations

Worked examples

Example 1 — a first encounter with Space diagonal

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

In research: Space diagonal appears in mathematics 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 Space diagonal 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: Space diagonal is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary geometry, Magic squares, so understanding it makes those chapters shorter.
In everyday life: Look for Space diagonal 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 Space diagonal in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Space diagonal 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 Space diagonal out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Space diagonal in simple terms?

In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the same face (but not on the same edge) as each other.

Why does Space diagonal matter?

Because it connects several mathematics 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 Space diagonal?

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 Space diagonal.