2024-02-11: Symbolic algebra and typing

2023-08-01: Population waves

2023-05-18: Math of telephone billing mystery

2023-05-05: Franklin and DNA More information…

2023-04-25: On angle and dimension

2023-02-20: On Leonardo da Vinci and Gravity

2022-04-29: Fabricating Evidence to catch Carmen Sandiego

2022-03-04: Probabilistic law of the excluded middle

2020-05-04: Archimedes and the sphere

2019-05-16: Glow worms return

2019-04-11: Original memetic sin

2019-01-31: The theory of weight

2018-11-06: Origins of telephone network theory

2018-10-24: Modern thought

2018-09-10: Feeding a controversy

2018-06-11: Glow worm distribution

2018-04-23: Outlawing risk

2017-08-22: A rebuttal on the beauty in applying math

2017-04-22: Free googles book library

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-02-17: Dependent Non-Commuting Random Variable Systems

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: Mathematical Epidemiology citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-12: A Comment on How Biased Dispersal can Preclude Competitive Exclusion

2015-02-09: Hamilton's selfish-herd paradox

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-17: Ebola comments

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why \(1^{\infty} eq 1\)

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the circle area formula using elementary methods

Blog – angle

On angle and dimension

One of the things I like about science is that it’s not frozen. It’s a stew of ideas, and when a better one comes along, it gets embraced over older ideas with worse flaws. Of course, that can be a relatively slow process compared to the speed of news in this day and age, and sometimes its more complicated, but we tend to get there more often than not.

So, I’ve been rather frustrated to discover that the basic theory of one of the core ideas in applied mathematics remains a mess. I speak of the subject of variables, types, units, and dimensions.

Variables are a necessary ingredient in mathematical and computational models. We cann’t build a model without them, and in some sense, we’re all experts – we understand estimation, precision, evaluation, representation, types, units, and dimensions. We understand these things practically at an intuitive level – we just know how things fit together without having to invoke an explicit theory. Yet, when we pay attention to the details, we find some things are rather incoherent. Terminology is poorly defined. Implementation is often over-looked. Variety seems to be swept under the rug, without serious concern.

Other authors have said similar things – in fact, it seems every serious effort on the topic of dimensional analysis spanning decades has said the same thing. And each has made valuable contributions. Yet, a basic and satisfactory classroom theory still seems to elude us.

Perhaps we shouldn’t care too much – certainly I didn’t until recently. Experts know what they’re doing without a need for such a theory. But that argument strikes me now as dark parallel to that of the “Practical men” of electrical engineering back before Maxwell’s laws were accepted.

Modern history of angle dimension

Angles are one spot where our theory seems confused. Here are some quotes from several authors on the topic.

A question may arise in connection with the dimensions of \(\theta\). We have said that it is dimensionless, and that its numerical magnitude does not change when the size of the fundamental units of mass, length, or time are changed. This of course is true, but it does not follow that therefore the numerical magnitude of is uniquely determined, as we see at once from the possibility of measuring in degrees or in radians. Are we therefore justified in treating as a constant and saying that it may enter the functional relation in any way whatever? – (Bridgman, 1922, page 3)

An angle, as can be seen from its expression in radians as the ratio of two lengths, is of zero dimensions. Hence the dimensions of angular velocity are \([T^{-1}]\); of angular acceleration, \([T^{-2}]\); and of angular momentum (moment of inertia x angular velocity), \([ML^2T^{-1}]\). (Pankhust, 1964, page 23)

(Langhaar, 1967) gives no discussion but requires angles to be dimensionless.

All angles are dimensionless (ratio of arc length to radius) and should be taken in radians for this reason. – (White, 1986, 249) - still in print in 2015.

The words radian and steradian merely serve as reminders of how the corresponding numbers were obtained. Nevertheless, there is no harm in including ‘radian’ or ‘steradian’ in any mathematical expression provided that we remember that we may put unity in their place whenever we wish. – (Massey 1986, page 37)

In 1995, the radian and steradian were abolished from the SI system of units, being deemed dimensionless and hence not needing official designation.

A dimension is a measure of a physical quantity (without numerical values), while a unit is a way to assign a number to that dimension. For example, length is a dimension that is measured in units such as microns (μm), feet (ft), centimeters (cm), meters (m), kilometers (km), etc. (Fig. 7-1). There are seven primary dimensions (also called fundamental or basic dimensions) … Dimensional variables are defined as dimensional quantities that change or vary in the problem. For the simple differential equation given in Eq. 7-4, there are two dimensional variables: z (dimension of length) and t (dimension of time). Nondimensional (or dimensionless) variables are defined as quantities that change or vary in the problem, but have no dimensions; an example is angle of rotation, measured in degrees or radians which are dimensionless units – (Cengal, 2006, pages 270 and 299)

On the other hand, there are quantities whose dimensions are 1 (one). These always occur when all the dimensions associated with the quantity cancel out. For example, the magnitude of a plane angle \(\varphi\) may be expressed in radians. But the radian (abbreviated “rad”) itself is the ratio of the length of the subtended arc \(\lambda\) of a circle and that circle’s radius R. Thus, we have \(\varphi = \lambda/R\), and hence we can write for the dimension of the angle \[[\varphi] = \text{rad} = \left[ \frac{\lambda}{R} \right] = \frac{[\lambda]}{[R]} = \frac{\text{meter}}{\text{meter}} = 1.\] Therefore, the dimension of the radian is 1. In this respect, a dimension of 1 is analogous to a coefficient of 1, or an exponent of 1. – (Szirtes, 2007, page 34)

Also included sometimes among the base quantities are two dimensionless quantities, plane angle and solid angle, which are measured in radians and steradians, respectively. We consider them derived quantities because, though dimensionless, they are defined in terms of operations involving length, much like area is defined in terms of length operations. – (Sonin 2001, page 30)

(Conejo, 2021) makes no mention of angles at all.

Even though frequency, angular velocity, angular frequency and radioactivity all have the dimension 1/T, of these only frequency is expressed in hertz. Thus a disc rotating at 60 revolutions per minute (rpm) is said to have an angular velocity of 2π rad/s and a frequency of rotation of 1 Hz. The correspondence between a frequency f with the unit hertz and an angular velocity ω with the unit radians per second is ω = 2πf ... – Wikipedia entry on Hertz in 2022

Current convention

What are we to make of this?

It was, in some sense, an abolishment of the identity of angles to a nether-world of non-dimensionlessness – existing in every practical manner, but the legal one. Angles have units (degree, radian, turn, grade), we often have to convert between units, they are directly measurable using a protractor, and they have a unique topology (\(-180^{\circ} = 180^{\circ}\)) not shared by other dimensionless ratios like aspect ratio.

It is commonly asserted in engineering and applied mathematics that angles are dimensionless quantities as Bridgman did a century ago. The argument is typically something likes this (I have changed symbol names for convenience) …

… On the other hand, there are quantities whose dimensions are 1 (one). These always occur when all the dimensions associated with the quantity cancel out. For example, the magnitude of a plane angle may be expressed in radians. But the radian (abbreviated “rad”) itself is the ratio of the length of the subtended arc \(p\) of a circle and that circle’s radius \(r\). Thus, we have \(\theta = p/r\), and hence we can write for the dimension of the angle

\[[\theta] = \text{rad} = \left[\frac{p}{r}\right] = \frac{[p]}{[r]} = \frac{\text{meter}}{\text{meter}} = 1\]

Therefore, the dimension of the radian is 1. …

Now, that’s … reasonable. There’s nothing formally wrong with it. But it’s not a proof. This argument only demonstrates consistency – if we measure arc \(p\) in meters and radius \(r\) in meters and define the angle \(\theta\) as the ratio of arc to radius, then angle must be dimensionless. However, it also viloates one of the guiding principles of dimensional analysis – to create formulas that are true independent of the units selected. The equation \(\theta = p/r\) is only true when we measure angles in radians! It is not true if we use degrees, or any other convention.

Richness of types

Under the dimensionless assertion, angles are no different than aspect ratios. This has inspired authors to remind readers that “… just because quantities have the same dimensions doesn’t mean the are the same things …” This usually arises when we discuss torques and frequencies, for which the SI dimensions do not identify the rotational component. Because, after all, angles are periodic, and one turn in either direction brings things back to their starting point. And that is something clearly not present in aspect ratios.

I contend that this type of hedging/waffling/dancing flaunts the very purpose of augmenting numbers with units and type systems. Units and dimensions were created to enrich numbers – to make it easier to communicate information and context and to avoid ambiguity. Is it really worth the ambiguity to have torques and energies measured in the same dimensions when they are naturally different things?

To insist that angles are “dimensionless” when they clearly are their own kind of thing, and can be measured directly with their own units in complete independent parallel with lengths? That seems like a classic misguided attempt to extend a framework beyond its natural scope.


There is another way to think of the relationship between angle and arc – as a conversion. Suppose we measure an angle between two bearings. That angle corresponds to an arc that is part of a circle, and that arc has a length. Formally, this should be calculated as an arc-length integral. Calling the arc a curve given by \(\gamma(t)\) for \(t \in [0,\theta]\), the arc-lenth formula from calculus is \[p = \int_{\theta_1}^{\theta_2} \left|\frac{d\gamma}{dt} \right| dt.\] Observe that this formula is independent of the choice of units of \(t\) because the local arclength is a derivative with units of distance per \(dt\). So, let’s take length in units of meters and angle (our integration parameter) in units of radians (rad). Applying with \(\gamma(t) = r e^{it}\), \(d\gamma/dt = r i e^{it}\) and \(| r i e^{it}| = r\) with units of meters per radian. Then \[\begin{align*} p = \int_{0}^{\theta} | r i e^{it}| dt = \int_{0}^{\theta} r dt = r \int_{0}^{\theta} dt = r \theta. \end{align*}\] where \(r\) still has units of meters per radian! \[[ p = r \theta] \rightarrow [ p ] = [ r ] [ \theta ] \rightarrow (\text{meter}) = (\text{meter}/\text{rad}) (\text{rad})\] If radians are dimensionless \(( \text{rad} = \text{1})\), then this reduces to the standard formula. However, radians need not be DIMENSIONLESS. The relationship holds even without this assumption. Thus, it is not necessary to treat radians as dimensionless as long as one treats the necessary conversions as lengths per unit of angle.

The length of the outer perimeter \(p\) of a sector of a circle with angle \(\theta\) \[\begin{align*} p \; \text{meters} &= (r \; \text{meters/radian}) &&(\theta \;\text{radians}) \\ &= (2 \pi r \; \text{meters/turn}) &&(\theta \;\text{turns}) \\ &= (\pi r /180 \; \text{meters/degree}) &&(\theta \;\text{degrees}) \\ &= (\pi r /200 \; \text{meters/grad}) &&(\theta \;\text{grads}) \end{align*}\]

or when we explicitly introduce the necessary conversion factors,

\[\begin{align*} p \; \text{meters} &= (r \; \text{meters}) (1 \; \text{per radian}) &&(\theta \;\text{radians}) \\ &= (r \; \text{meters}) (2 \pi \; \text{per turn}) &&(\theta \;\text{turns}) \\ &= (r \; \text{meters}) (\pi /180 \; \text{per degree}) &&(\theta \;\text{degrees}) \\ &= (r \; \text{meters}) (\pi /200 \; \text{per grad}) &&(\theta \;\text{grads}) \end{align*}\] where \[ (2 \pi \;\text{radians}) =(1 \;\text{turn}) =(360 \;\text{degrees}) =(400 \;\text{grads}). \]

\[ p = r f \theta \] where \(f\) is the angular frequency = the circumference of a unit circle per unit radius divided by the number of angle units in a unit circle, with net dimensions of reciprocal angle.

Topologies of dimensions

When I look at scalars – continuous one-dimensional variables –, I see 6 natural situations that your system of variable dimensions should acknowledge. (I’ve never seen this picture anyplace before, but perhaps somebody knows a reference.)

Dimensional analysis only applies to these first two.

These next four do not have the additivity necessary for dimensional analysis.

I’m joining a certain conversation late, here, I now see. But I seem to be in general agreement with Angles in the SI: a detailed proposal for solving the problem by Quincey, Paul. Metrologia, 2021, v. 58 (5), pp. 053002. DOI:10.1088/1681-7575/ac023f