Scale of the Universe Explorer

zoom through powers of ten, from quantum length to cosmic horizon

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Lesson

The theory — Scale of the Universe Explorer

An order of magnitude is one factor of ten. Write a length as 10^a metres and its size collapses into a single number, a — and equal steps across the ruler above are equal multiplications, not equal distances.

What each symbol means

a: the exponent in 10^a m — the only number you need to place something on this ruler. A bacterium is 10⁻⁶ m, so its a is −6; Everest is 10⁴ m, so its a is 4.
N:M: a ratio, written so the larger object comes first and the smaller is scaled to 1. Bacterium to virus is 10¹ : 1 — the bacterium is ten times the length.
decade: one factor of ten, and the unit the ratio table counts in. The gap between two objects is a − b decades, so a difference of 3 decades means a thousandfold, not threefold.

Where the formula comes from

To compare two sizes you divide them: 10^a metres by 10^b metres.
Dividing powers of the same base subtracts the exponents, so 10^a / 10^b = 10^a−b. The division itself never has to be carried out — and that is the whole reason a logarithmic ruler is usable at these extremes.
The ratio is therefore 10^a−b : 1, and a − b is the number of decades between the two. That subtraction is what the table above prints: the Sun's radius against Everest is 8.847 − 4.000 = 4.847 decades.

How to read what you see

Drag to pan, and scroll or pinch to zoom; the ruler is marked in powers of ten, so one step is one factor of ten wherever you are on it. The table lists pairs of objects three ways at once — the two names, the ratio as N:M, and the exponent subtraction that produced it. Reading a row backwards is often the more surprising direction: a bacterium against an atom is 10⁴ : 1, so an atom is ten thousand times shorter than a single cell of the simplest life there is.

Assumes: One representative length per object, which is a genuine simplification rather than a rounding: bacteria span more than a decade among themselves, and “human cell” covers both a red blood cell and a neuron a metre long. The ruler also measures length only. Mass, age and energy each span their own, quite different, number of decades.
Breaks when: Two objects inside the same decade land almost on top of each other, so this is the wrong instrument for comparing things of similar size — it is built for exactly the comparisons a linear ruler cannot show. And the smallest label is not a size at all: the Planck length is not the width of any object, it is the scale at which our current description of space stops giving reliable answers.

why a logarithmic universe map works 🖖

A linear ruler cannot show a bacterium, a mountain, a star, and a galaxy on the same screen. A logarithmic ruler can, because equal visual steps mean equal multiplication factors. Subtracting exponents turns impossible-looking scale comparisons into simple ratio arithmetic: 10^9 m divided by 10^4 m is 10^5:1.

every step multiplies by ten 🖖

This tool is really a journey through the powers of ten: each notch on the ruler does not add a fixed distance, it multiplies size by 10. That is why dragging just a few finger-widths leaps past a bacterium, a whale, and a planet. From the Planck length (10⁻³⁵ m) to the observable universe (10²⁶ m) is about 61 such steps — written out in full, that ratio would need 61 zeros.

life sits near the middle of everything 🖖

Walk exactly halfway along this ruler and you do not land on human size — you land on the geometric mean of the smallest and largest lengths, about 3 × 10⁻⁵ m. That is a few tens of micrometres: the scale of a living cell or a fine human hair. So on an honest logarithmic map of all sizes, cellular life sits astonishingly close to the true midpoint between the Planck length and the cosmic horizon.

Problem solved in full

The ratio of the Sun's and Moon's mean distances by subtracting exponents 6 steps
The Moon's mean distance is 3.844×10⁸ m and the Sun's is 1.4960×10¹¹ m. Get the ratio by subtracting exponents instead of dividing — then use it to predict how many times larger the Sun is than the Moon, and work out why the prediction misses.
1. Two measured means, one per orbit. Neither number comes off the ruler above; the ruler was built from them.
  
  $d_{M} = 3.844 \times 1 0^{8} m, d_{S} = 1.4960 \times 1 0^{11} m$
2. Divide these as written and you are pushing twelve digits around. Put each distance on the exponent instead — a number written as 10^x keeps all of its size in x. Those two values, rounded to three decimals, are exactly the positions the ruler holds for the Moon's and the Sun's distance.
  
  $lo g_{10} d_{M} = 8 + lo g_{10} 3.844 = 8.5848, lo g_{10} d_{S} = 11 + lo g_{10} 1.4960 = 11.1749$
3. Powers of ten divide by subtracting. Click the Earth-Sun distance and Earth-Moon distance labels on the ruler — this tool has no preset buttons, the labels themselves are the control. The Exponent math column then prints this subtraction, and the N:M column the ratio it stands for.
  
  $\frac{d _{S}}{d _{M}} = \frac{1 0 ^{11.175}}{1 0 ^{8.585}} = 1 0^{11.175 - 8.585} = 1 0^{2.590}$
4. An exponent is not yet a ratio. Split it — the whole part 2 counts the decades, the leftover 0.590 supplies the leading digits, because 10^0.590 = 3.890. Dividing the two original distances the long way gives 389.18, and agreeing to three figures is the receipt that the trip through logarithms lost nothing.
  
  $1 0^{2.590} = 1 0^{2} \times 1 0^{0.590} = 100 \times 3.890 = 3.89 \times 1 0^{2}, \frac{1.4960 \times 1 0 ^{11}}{3.844 \times 1 0 ^{8}} = 389.18$
5. Now spend the ratio on something it was not asked for. From here the Sun and Moon look almost the same size, and a disc's angular size is roughly its diameter over its distance — so if the two discs matched exactly, the radius ratio would be forced to equal the distance ratio. Measured radii disagree: 400.43 against the 389.18 the distances demand, larger by 2.9%.
  
  $θ \approx \frac{2 R}{d}; θ_{S} = θ_{M} \Rightarrow \frac{R _{S}}{R _{M}} = \frac{d _{S}}{d _{M}} = 389, \frac{6.957 \times 1 0 ^{5} km}{1.7374 \times 1 0 ^{3} km} = 400.43$
6. The 2.9% is not an arithmetic slip. It is the two discs failing to match, and the fix is to stop assuming they do: put the radii and distances into the angle itself. The Moon is the smaller disc on average, which is not the way the coincidence is usually told.
  
  $θ_{S} = 2 arctan \frac{6.957 \times 1 0 ^{5} km}{1.4960 \times 1 0 ^{8} km} = 31.9 7^{'}, θ_{M} = 2 arctan \frac{1.7374 \times 1 0 ^{3} km}{3.844 \times 1 0 ^{5} km} = 31.0 8^{'}$
Answer

2.590 decades — a distance ratio of 3.89×10² : 1. The radius ratio is 400.43, larger by 2.9%, so on average the Moon is the smaller disc. That average is not what decides an eclipse. The Moon's distance swings between 363,300 km and 405,500 km, so its disc swings from 32.88′ down to 29.46′, while the Sun's moves only between 32.52′ and 31.45′. The two ranges overlap. Near perigee the Moon's disc is the larger one and covers the Sun completely — a total eclipse; near apogee it is too small and leaves a ring of Sun around it — an annular eclipse. The 2.590 decades is what makes the two discs agree to within 3% at all; the wobble around it decides which eclipse you get, and the ruler above computes none of that.

$\frac{d _{S}}{d _{M}} = 1 0^{2.590} = 3.89 \times 1 0^{2} \neq = \frac{R _{S}}{R _{M}} = 400.43 ⟹ θ_{M} < θ_{S}$

Learning path

Orders of magnitude

Leads to Fermi estimates the ruler with things on it, from a proton to the observable universe, every step a factor of ten.

References (2)

The Planck length, as measured — 1.616 255 × 10⁻³⁵ m, which the page rounds to 10⁻³⁵ m: NIST/CODATA, “Planck length”, Fundamental Physical Constants.
The scale of the observable universe comes from the cosmological parameters here: Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters.” Astronomy & Astrophysics 641, A6, 2020.

By the 42Math team · Updated 2026-05-09

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Quantum - Quantum realm: Planck length (10⁻³⁵ m) through atomic nucleus to atom (10⁻¹⁰ m)
Human - Human scale: bacterium (10⁻⁶ m) through ant, human (1 m) to mountain (10⁴ m)
Solar - Solar system: Earth radius through Earth–Sun distance (1 AU ≈ 10¹¹ m)
Cosmic - Cosmic scale: light-year through Milky Way to observable universe (10²⁶ m)
Sun vs Moon - The Sun is 3.89×10² times further away than the Moon, and the ruler gets there by subtracting exponents: 11.175 − 8.585 = 2.590 decades.