equal-parts-without-diameter

Equal Parts Without a Diameter

Summary: A curve can have $n$ similar and equal parts arranged with $n$-fold rotational symmetry about a center $C$ but no diameter — no axis of reflection (§§355–360). The construction parallels n-diameters-by-cos-ns except that the polar invariant must now be invariant under $s\mapsto \frac{2\pi }{n}+s$ (rotation by $2\pi /n$) without also being invariant under $s\mapsto -s$ (reflection). Both $\cos ns$ and $\sin ns$ satisfy the rotation condition, so the curve equation involves both the “Chebyshev real part” $\cos ns$ and the “Chebyshev imaginary part” $\sin ns$. In rectangular coordinates, the latter appears as the imaginary part of $(x+iy{)}^n$: $2xy$ for $n=2$, $3x^{2}y-y^3$ for $n=3$, $4x^{3}y-4x{y}^3$ for $n=4$, $5x^{4}y-10x^2{y}^3+y^5$ for $n=5$, etc. Dropping the $\sin ns$ part recovers the diameter case.

Sources: chapter15 (§§355–360); figures 73 ($n=2$ with no diameter), 74 ($n=3$ with no diameter) in figures72-75.

Last updated: 2026-05-11.

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How equal-part counts work (§355)

A curve with $n$ diameters has $2n$ similar and equal parts: each diameter divides the curve into two halves, and the $n$ diameters together break the curve into $2n$ alternating slices. Examples:

Diameters	Parts	Figure
2	4 ($AE,BE,AF,BF$)	70
3	6 ($AE,GE,GB,FB,FH,AH$)	71
4	8 ($AE,AK,GE,GI,BI,BF,HF,HK$)	72

But — and this is the new observation of §355 — a curve can have multiple equal parts without having any diameter at all. The minimum number is $2$ (one rotational pair). The number of equal parts is always at least twice the number of diameters, but can exceed that, by exactly the count of rotational-only symmetries that have no companion reflection.

Two opposite parts, no diameter (§§356–357, figure 73)

Take two equal parts $AME$ and $BKF$ situated in opposite regions. Setting $z=CM,s=\angle ACM$ and using the rotational pairing $s\leftrightarrow \pi +s$ (i.e., $z$ takes the same value at $M$ and at the diametrically-opposite point $K$ with $CK=CM$):

$z(s)=z(\pi +s).$

Expressions invariant under this — but not under $s\mapsto -s$ — include $\tan s$ (since $\tan s=\tan (\pi +s)$). Equivalently, working with sines and cosines to avoid fractions: $\sin 2s=\sin 2(\pi +s),\cos 2s=\cos 2(\pi +s).$ $Z$ is a non-irrational function of $z$, $\sin 2s$, and $\cos 2s$ — i.e., in rectangular coordinates, of $x^2+y^2$, $\cos 2s=(x^2-y^2)/(x^2+y^2)$, and $\sin 2s=2xy/(x^2+y^2)$. Cleanly: $Z$ is a non-irrational function of $x^2$, $y^2$, and $xy$. The general equation is

$0=\alpha +\beta x^2+\gamma xy+\delta y^2+ϵx^4+\zeta x^{3}y+\eta x^2{y}^2+\theta x{y}^3+\iota y^4+\cdots .$

This is the general §340 (even-degree homogeneous parts) form — a center but possibly no diameter. If the $xy$ term (and its higher-degree analogs) vanish, $Z$ becomes a function of $x^2,y^2$ alone, and the curve recovers the two perpendicular diameters of §342. Hence the ”$\sin 2s$ part” is exactly the obstruction to having diameters.

Three parts at $120°$, no diameter (§358, figure 74)

Three equal parts $AM,BN,DL$ arranged with $3$-fold rotational symmetry about $C$, with the three “spokes” $CM,CN,CL$ at angles $s,\frac{2\pi }{3}+s,\frac{4\pi }{3}+s$ all giving the same value of $z$. Invariants under $s\mapsto \frac{2\pi }{3}+s$ are functions of $\sin 3s$ and $\cos 3s$: $\cos 3s=\frac{x^3-3x{y}^2}{z^3},\sin 3s=\frac{3x^{2}y-y^3}{z^3}.$ So $Z$ is a non-irrational function of $x^2+y^2$, $3x^{2}y-y^3$, and $x^3-3x{y}^2$. The general equation:

$0=\alpha +\beta (x^2+y^2)+\gamma (3x^{2}y-y^3)+\delta (x^3-3x{y}^2)+ϵ(x^2+y^2{)}^2+\cdots .$

A third-order line with three equal parts but no diameter has the form $0=\alpha +\beta x^2+\beta y^2+\delta x^3+3\gamma x^{2}y-3\delta x{y}^2-\gamma y^3.$

If $\gamma =0$ (the $\sin 3s$ part vanishes), this collapses to the §351 diameter form $x^3-3x{y}^2=a(x^2+y^2)+b^3$.

Four parts at $90°$, no diameter (§359, figure 73 reused for analogous geometry)

Four equal parts arranged with $4$-fold rotational symmetry about $C$. Invariants under $s\mapsto \frac{\pi }{2}+s$: $\sin 4s=\frac{4x^{3}y-4x{y}^3}{z^4},\cos 4s=\frac{x^4-6x^2{y}^2+y^4}{z^4}.$ $Z$ is a non-irrational function of $x^2+y^2$, $4x^{3}y-4x{y}^3$, and $x^4-6x^2{y}^2+y^4$. The general equation:

$0=\alpha +\beta x^2+\beta y^2+\gamma x^4+\delta x^{3}y+ϵx^2{y}^2-\delta x{y}^3+\gamma y^4+\cdots ,$

with the constraint that the quartic-degree pieces are an $\mathbb{R}$-linear combination of $4x^{3}y-4x{y}^3$ and $x^4-6x^2{y}^2+y^4$ (cf. §359 textually).

General $n$ equal parts no diameter (§360)

For $n$ equal parts at angle $2\pi /n$, $Z$ is a non-irrational function of $x^2+y^2$ and the two Chebyshev polynomials of degree $n$ — the real and imaginary parts of $(x+iy{)}^n$. Explicitly:

$\sin ns/z^n=\left(\genfrac{}{}{0ex}{}{n}{1}\right)x^{n-1}y-\left(\genfrac{}{}{0ex}{}{n}{3}\right)x^{n-3}{y}^3+\left(\genfrac{}{}{0ex}{}{n}{5}\right)x^{n-5}{y}^5-\cdots$

(the imaginary part), and

$\cos ns/z^n=x^n-\left(\genfrac{}{}{0ex}{}{n}{2}\right)x^{n-2}{y}^2+\left(\genfrac{}{}{0ex}{}{n}{4}\right)x^{n-4}{y}^4-\cdots$

(the real part, the Chebyshev real part). The closing observation:

If either of these last two expressions are missing, then the curve has $n$ diameters. (source: chapter15, §360)

Dropping $\sin ns$ recovers the $n$-diameter case; dropping $\cos ns$ also gives a curve with $n$ diameters, but reflected by $\pi /(2n)$ — the diameters now bisect the gaps between where the original ones would have been.

Symmetry, in summary

Polar invariant	Geometric meaning
$z=f(\cos s)$ only	one diameter
$z=f(\cos 2s)$ only	two perpendicular diameters
$z=f(\cos ns)$ only	$n$ diameters at angle $\pi /n$
$z=f(\cos ns,\sin ns)$ both	$n$-fold rotational symmetry, no diameters
$z=$ any	no symmetry

The role of the imaginary part $\sin ns$ is precisely to break the reflective symmetry while preserving the rotational symmetry — turning diametrically equal parts into alternately equal ones.

Figures

Figures 72–75

Related pages

• chapter-15-on-curves-with-one-or-several-diameters
• diameter-and-center-from-equation — the $n=1,2$ counterpart with reflective symmetry.
• n-diameters-by-cos-ns — the $\cos ns$-only construction; equal parts arrange around diameters.
• concurrence-of-diameters — establishes the $\pi /n$ spacing.
• classification-of-equal-parts — the §§361–363 completeness theorem covering both this and the diameter case.