3730:
27:
4481:
4606:
601:
3340:
3725:{\displaystyle {\begin{aligned}{\frac {1}{2}}\int _{0}^{2\pi }(a\cos(k\theta ))^{2}\,d\theta &={\frac {a^{2}}{2}}\left(\pi +{\frac {\sin(4k\pi )}{4k}}\right)={\frac {\pi a^{2}}{2}}&&\quad {\text{for even }}k\\{\frac {1}{2}}\int _{0}^{\pi }(a\cos(k\theta ))^{2}\,d\theta &={\frac {a^{2}}{2}}\left({\frac {\pi }{2}}+{\frac {\sin(2k\pi )}{4k}}\right)={\frac {\pi a^{2}}{4}}&&\quad {\text{for odd }}k\end{aligned}}}
1257:
1208:
4649:
4528:
1540:. These rose's positive and negative half-cycles are coincident, which means that in graphing them, only the positive half-cycles or only the negative half-cycles need to plotted in order to form the full curve. (Equivalently, a complete curve will be graphed by plotting any continuous interval of polar angles that is
877:) the plot of a half-cycle can be seen as spiraling out from the pole in more than one circuit around the pole until plotting reaches the inscribed circle where it spirals back to the pole, intersecting itself and forming one or more loops along the way. Consequently, each petal forms two loops when
354:
486:
3126:
4350:
3283:
2505:
1850:
1753:
2928:
2794:
2630:
1188:
Individual petals are symmetric about the line through the pole and the petal's peak, which reflects the symmetry of the half-cycle of the underlying sinusoid. Roses composed of a finite number of petals are, by definition,
1996:
196:
2333:
2225:
2091:
365:
3345:
201:
3970:
are coincident. For such a pair of roses, the rose with the sine function specification is coincident with the crest of the rose with the cosine specification at on the polar axis either at
1661:. The circle is the curve's single petal. (See the circle being formed at the end of the next section.) In Cartesian coordinates, the equivalent cosine and sine specifications are
185:
2969:
4216:
3925:
are odd, the positive and negative half-cycles of the sinusoid are coincident. The graph of these roses are completed in any continuous interval of polar angles that is
786:
Consistent with the rules for plotting points in polar coordinates, a point in a negative half-cycle cannot be plotted at its polar angle because its radial coordinate
3137:
1564:
The roses are symmetric about each line through the pole and a peak (through the middle of a petal) with the polar angle between the peaks of successive petals being
1450:
The roses are symmetric about each line through the pole and a peak (through the "middle" a petal) with the polar angle between the peaks of successive petals being
2374:
80:
are rotations of these roses by one-quarter period of the sinusoid in a counter-clockwise direction about the pole (origin). For proper mathematical analysis,
1764:
1667:
2805:
2671:
621:. A petal is the shape formed by the graph of a half-cycle of the sinusoid that specifies the rose. (A cycle is a portion of a sinusoid that is one period
2516:
1891:
705:
The shape of each petal is same because the graphs of half-cycles have the same shape. The shape is given by the positive half-cycle with crest at
349:{\displaystyle {\begin{aligned}x&=r\cos(\theta )=a\cos(k\theta )\cos(\theta )\\y&=r\sin(\theta )=a\cos(k\theta )\sin(\theta )\end{aligned}}}
2949:
2236:
2128:
2651:
4786:
with a crest on the polar axis; however there is no other polar angle in the domain of the polar equation that will plot at the coordinates
1504:
The roses are symmetric about each line that bisects the angle between successive peaks, which corresponds to half-cycle boundaries and the
2354:
5300:
2007:
481:{\displaystyle \sin(k\theta )=\cos \left(k\theta -{\frac {\pi }{2}}\right)=\cos \left(k\left(\theta -{\frac {\pi }{2k}}\right)\right)}
804:. Thus, positive and negative half-cycles can be coincident in the graph of a rose. In addition, roses are inscribed in the circle
5291:
4921:
4903:
1252:
the octagon makes sketching the graph relatively easy after the half-cycle boundaries (corresponding to apothems) are drawn.
1561:.) Line segments connecting successive peaks will form a regular polygon with an odd number of vertices, and likewise:
2122:. The curve is also called the Paquerette de Mélibée. In Cartesian Coordinates the cosine and sine specifications are
4690:) is each, a single loop that intersect other petals. The rose is symmetric about the pole. The rose is complete at
828:, the petal's shape is a single closed loop. A single loop is formed because the angle interval for a polar plot is
4397:
curves that can be used to trisect angles. The rose has a single petal with two loops. (See the animation below.)
1090:
3297:
1444:
with an even number of vertices that has its center at the pole and a radius through each peak, and likewise:
775:
of this petal about the pole, including those for roses specified by the sine function with same values for
553:
3121:{\displaystyle \left(x^{2}+y^{2}\right)^{7}=a^{2}\left(x^{6}-15x^{4}y^{2}+15x^{2}y^{4}-y^{6}\right)^{2}}
5243:
146:
4794:. Overall, roses specified by sinusoids with angular frequencies that are irrational constants form a
1390:
cycles displayed in the graph. No additional points need be plotted because the radial coordinate at
1193:
since each petal is the same shape with successive petals rotated about the same angle about the pole.
5026:
1407:(which are crests for two different positive half-cycles for roses specified by the cosine function).
3951:
long. Furthermore, the roses are symmetric about the pole for both cosine and sine specifications.
4908:
4345:{\displaystyle \left(x^{2}+y^{2}\right)\left(2\left(x^{2}+y^{2}\right)-a^{2}\right)^{2}=a^{4}x^{2}}
549:
3944:
is odd, or visa versa, the rose will be completely graphed in a continuous polar angle interval
3278:{\displaystyle \left(x^{2}+y^{2}\right)^{7}=4a^{2}\left(3x^{5}y-10x^{3}y^{3}+3xy^{5}\right)^{2}}
5201:
5180:
4981:
1190:
5071:
5327:
4898:
4817:
4534:
4390:
1333:
is odd. The properties of these roses are a special case of roses with angular frequencies
4868:
3795:
2119:
548:
Since they are specified using the cosine or sine function, roses are usually expressed as
2500:{\displaystyle \left(x^{2}+y^{2}\right)^{5}=a^{2}\left(x^{4}-6x^{2}y^{2}+y^{4}\right)^{2}}
8:
5159:
5117:
5092:
5051:
4960:
4894:
1530:
interval of polar angles displayed. Each peak corresponds to a point lying on the circle
1430:
interval of polar angles displayed. Each peak corresponds to a point lying on the circle
588:, a rose curve can be expressed in Cartesian coordinates since those can be specified as
26:
4648:
4605:
4527:
4480:
3962:
is even, roses specified by the cosine and sine polar equations with the same values of
1845:{\displaystyle x^{2}+\left(y-{\frac {a}{2}}\right)^{2}=\left({\frac {a}{2}}\right)^{2}}
1748:{\displaystyle \left(x-{\frac {a}{2}}\right)^{2}+y^{2}=\left({\frac {a}{2}}\right)^{2}}
120:. Rose curves or "rhodonea" were named by the Italian mathematician who studied them,
4718:
2923:{\displaystyle \left(x^{2}+y^{2}\right)^{3}=a\left(5x^{4}y-10x^{2}y^{3}+y^{5}\right)}
2789:{\displaystyle \left(x^{2}+y^{2}\right)^{3}=a\left(x^{5}-10x^{3}y^{2}+5xy^{4}\right)}
557:
117:
4725:
has an infinite number of petals and will never complete. For example, the sinusoid
4102:
5222:
5138:
2662:
2625:{\displaystyle \left(x^{2}+y^{2}\right)^{5}=16a^{2}\left(xy^{3}-yx^{3}\right)^{2}}
1060:
5306:
4863:
3832:
are non-zero integers, the number of petals is the denominator of the expression
608:
Roses are directly related to the properties of the sinusoids that specify them.
589:
585:
20:
2338:
respectively. (See the trifolium being formed at the end of the next section.)
648:
long and consists of a positive half-cycle, the continuous set of points where
600:
138:
5311:
1991:{\displaystyle \left(x^{2}+y^{2}\right)^{3}=a^{2}\left(x^{2}-y^{2}\right)^{2}}
5321:
113:
4798:(that is, they come arbitrarily close to specifying every point in the disk
4926:
4845:
1871:
121:
5264:
5002:
4939:
4858:
1356:
Because a polar coordinate plot is limited to polar angles between 0 and
1337:
that are rational numbers discussed in the next section of this article.
1044:
A rose's petals will not intersect each other when the angular frequency
89:
4929:
and A.P. Rollett, second edition, 1961 (Oxford
University Press), p. 73.
4101:
is called the Dürer folium, named after the German painter and engraver
2328:{\displaystyle \left(x^{2}+y^{2}\right)^{2}=a\left(3x^{2}y-y^{3}\right)}
2220:{\displaystyle \left(x^{2}+y^{2}\right)^{2}=a\left(x^{3}-3xy^{2}\right)}
4873:
4394:
4356:
1646:
that lies on the pole with a diameter that lies on the polar axis when
987:), etc. Roses with only one petal with multiple loops are observed for
1063:
due to the underlying symmetric and periodic properties of sinusoids.
4795:
2960:
1285:
petals. Line segments connecting successive peaks lie on the circle
1178:
that makes the roses specified by the two polar equations coincident.
1111:
that makes the roses specified by the two polar equations coincident.
771:). The petal is symmetric about the polar axis. All other petals are
565:
101:
1237:
petals. Line segments connecting successive peaks lie on the circle
1293:
772:
2963:. In Cartesian Coordinates the cosine and sine specifications are
2665:. In Cartesian Coordinates the cosine and sine specifications are
2368:. In Cartesian Coordinates the cosine and sine specifications are
2086:{\displaystyle \left(x^{2}+y^{2}\right)^{3}=4\left(axy\right)^{2}}
1885:. In Cartesian coordinates the cosine and sine specifications are
137:
A rose is the set of points in polar coordinates specified by the
2365:
1505:
1441:
1440:. Line segments connecting successive peaks will form a regular
1245:
835:
and the angular width of the half-cycle is less than or equal to
4487:
1882:
1643:
1256:
1207:
1048:
is a non-zero integer; otherwise, petals intersect one another.
105:
41:
for various rational numbered values of the angular frequency
4070:, corresponding to the radial coordinate of all of its peaks.
1587:
radians. Thus, these roses have rotational symmetry of order
1494:
radians. Thus, these roses have rotational symmetry of order
1351:, corresponding to the radial coordinate of all of its peaks.
604:
Artistic depiction of roses with different parameter settings
545:
radians, which is one-quarter the period of either sinusoid.
4707:
4573:), has one petal with two loops. The rose is complete when
359:
Roses can also be specified using the sine function. Since
190:
or in
Cartesian coordinates using the parametric equations
109:
3780:
1311:
is a non-zero integer, the curve will be rose-shaped with
1197:
1603:
The roses can be specified by algebraic curves of order
695:
long, and a negative half-cycle is the other half where
4219:
3343:
3140:
2972:
2808:
2674:
2519:
2377:
2239:
2131:
2010:
1894:
1767:
1670:
1041:, etc. (See the figure in the introduction section.)
368:
199:
149:
4210:. In Cartesian coordinates the rose is specified as
794:
radians to the polar angle with a radial coordinate
4642:
is reached (half a revolution of the lighter gear).
4521:
is reached (half a revolution of the lighter gear).
4344:
3724:
3277:
3120:
2922:
2788:
2624:
2499:
2327:
2219:
2085:
1990:
1844:
1747:
480:
348:
179:
4892:
4747:, so, it has a petal in the polar angle interval
5319:
1416:is even (and non-zero), the rose is composed of
1183:Only certain roses are symmetric about the pole.
5027:"Number of Petals of Odd Index Rhodonea Curve"
4359:, a curve that can be used to trisect angles.
1523:petals, one for each crest (or trough) in the
5262:
5090:
5000:
790:is negative. The point is plotted by adding
1082:is symmetric about the polar axis (the line
5241:
5157:
5115:
5049:
4958:
4416:The rays displayed are the polar axis and
4414:created using gears with different ratios.
3894:. This means that the number of petals is
3300:of a rose with polar equation of the form
5301:Visual Dictionary of Special Plane Curves
5220:
5136:
4362:
3592:
3410:
821:of the sinusoid is less than or equal to
4708:Roses with irrational number values for
3291:
1255:
1206:
599:
25:
5312:Create a rose curve as a vector graphic
4904:MacTutor History of Mathematics Archive
4700:(five revolutions of the lighter gear).
1447:The roses are symmetric about the pole.
1059:All roses display one or more forms of
728:(that is bounded by the angle interval
5320:
4820:- has the same shape as the rose with
3781:Roses with rational number values for
1296:. The rose is inscribed in the circle
1198:Roses with non-zero integer values of
84:must be expressed in irreducible form.
4466:otherwise, and proceeds clockwise to
3754:petals, so the area of each petal is
2111:is called a trifolium because it has
1131:is symmetric about the vertical line
595:
572:that determine the radial coordinate
4060:The rose is inscribed in the circle
1519:is odd, the rose is composed of the
1341:The rose is inscribed in the circle
5199:
5178:
4979:
4074:
2936:
1858:
127:
124:, between the years 1723 and 1728.
13:
2638:
507:is identical to that specified by
14:
5339:
5307:Interactive example with JSXGraph
5285:
4599:revolutions of the lighter gear).
2341:
1598:The rose’s petals do not overlap.
1423:petals, one for each peak in the
180:{\displaystyle r=a\cos(k\theta )}
4647:
4604:
4526:
4479:
4049:with non-zero integer values of
2099:
1229:is an even number, the rose has
617:Graphs of roses are composed of
556:) graphs of sinusoids that have
132:
30:Roses specified by the sinusoid
5256:
5235:
5214:
5193:
5172:
5151:
5130:
5109:
5084:
4717:A rose curve specified with an
3709:
3520:
1278:is an odd number, the rose has
19:For the topological usage, see
5064:
5043:
5019:
4994:
4973:
4952:
4932:
4914:
4886:
4654:The 8 petals of the rose with
4511:). The rose is complete when
3662:
3650:
3583:
3579:
3570:
3558:
3473:
3461:
3401:
3397:
3388:
3376:
384:
375:
339:
333:
324:
315:
300:
294:
268:
262:
253:
244:
229:
223:
174:
165:
1:
4632:). The rose is complete when
1630:
1508:of the corresponding polygon.
522:rotated counter-clockwise by
3794:is a rational number in the
492:Thus, the rose specified by
7:
5292:Applet to create rose with
4811:
4355:The Dürer folium is also a
4163:are coincident even though
1054:
10:
5344:
3334:is a non-zero integer, is
18:
5314:(using the sine function)
4393:that has the property of
4105:. The roses specified by
4019:. (This means that roses
611:
4909:University of St Andrews
4879:
2364:petals and will form an
2118:petals and will form an
1155:because of the identity
104:specified by either the
5093:"Paquerette de Mélibée"
2959:petals and will form a
2661:petals and will form a
1881:petals and will form a
1248:. Since one peak is at
16:Multi-lobed plane curve
4363:The limaçon trisectrix
4346:
4053:are never coincident.)
3917:In the case when both
3726:
3279:
3122:
2924:
2790:
2626:
2501:
2329:
2221:
2087:
1992:
1846:
1749:
1304:
1260:The rose specified by
1253:
1191:rotationally symmetric
605:
576:given the polar angle
482:
350:
181:
85:
4848:– a rose curve where
4347:
3727:
3292:Total and petal areas
3280:
3123:
2925:
2791:
2627:
2502:
2330:
2222:
2088:
1993:
1847:
1750:
1544:radians long such as
1397:is the same value at
1259:
1210:
603:
483:
351:
182:
29:
5265:"Rose (Mathematics)"
5003:"Rose (Mathematics)"
4940:"Rose (Mathematics)"
4895:Robertson, Edmund F.
4869:Sectrix of Maclaurin
4217:
3796:irreducible fraction
3341:
3138:
2970:
2806:
2672:
2517:
2375:
2237:
2129:
2120:equilateral triangle
2008:
1892:
1765:
1668:
1116:A rose specified as
1067:A rose specified as
932:), three loops when
554:Cartesian coordinate
366:
197:
147:
5263:Eric W. Weisstein.
5091:Eric W. Weisstein.
5001:Eric W. Weisstein.
4922:Mathematical Models
4893:O'Connor, John J.;
4442:Graphing starts at
3739:is even, there are
3557:
3375:
116:that is plotted in
69:Roses specified by
4818:Limaçon trisectrix
4535:limaçon trisectrix
4403:Examples of roses
4391:limaçon trisectrix
4342:
3954:In addition, when
3750:is odd, there are
3722:
3720:
3543:
3358:
3275:
3118:
2920:
2786:
2622:
2497:
2325:
2217:
2083:
1988:
1842:
1745:
1305:
1254:
606:
596:General properties
478:
346:
344:
177:
112:functions with no
86:
5269:Wolfram MathWorld
5097:Wolfram MathWorld
5007:Wolfram MathWorld
4719:irrational number
3790:In general, when
3746:petals; and when
3713:
3704:
3674:
3636:
3621:
3541:
3524:
3515:
3485:
3439:
3356:
1830:
1801:
1733:
1691:
1244:and will form an
1089:) because of the
558:angular frequency
466:
418:
118:polar coordinates
5335:
5295:
5279:
5278:
5276:
5275:
5260:
5254:
5253:
5251:
5250:
5242:Robert Ferreol.
5239:
5233:
5232:
5230:
5229:
5218:
5212:
5211:
5209:
5208:
5197:
5191:
5190:
5188:
5187:
5176:
5170:
5169:
5167:
5166:
5158:Robert Ferreol.
5155:
5149:
5148:
5146:
5145:
5134:
5128:
5127:
5125:
5124:
5116:Robert Ferreol.
5113:
5107:
5106:
5104:
5103:
5088:
5082:
5081:
5079:
5078:
5068:
5062:
5061:
5059:
5058:
5050:Robert Ferreol.
5047:
5041:
5040:
5038:
5037:
5023:
5017:
5016:
5014:
5013:
4998:
4992:
4991:
4989:
4988:
4977:
4971:
4970:
4968:
4967:
4959:Robert Ferreol.
4956:
4950:
4949:
4947:
4946:
4936:
4930:
4918:
4912:
4911:
4890:
4854:
4841:
4840:
4838:
4837:
4834:
4831:
4807:
4793:
4785:
4784:
4782:
4781:
4778:
4775:
4764:
4762:
4761:
4758:
4755:
4746:
4739:
4724:
4712:
4699:
4689:
4682:
4675:
4674:
4672:
4671:
4668:
4665:
4651:
4641:
4631:
4624:
4617:
4608:
4598:
4596:
4595:
4592:
4589:
4582:
4572:
4565:
4558:
4557:
4555:
4554:
4551:
4548:
4530:
4520:
4510:
4503:
4496:
4483:
4472:
4465:
4455:
4451:
4439:
4438:
4436:
4435:
4432:
4429:
4413:
4388:
4387:
4385:
4384:
4381:
4378:
4351:
4349:
4348:
4343:
4341:
4340:
4331:
4330:
4318:
4317:
4312:
4308:
4307:
4306:
4294:
4290:
4289:
4288:
4276:
4275:
4252:
4248:
4247:
4246:
4234:
4233:
4209:
4207:
4205:
4204:
4201:
4198:
4185:
4183:
4182:
4179:
4176:
4162:
4160:
4158:
4157:
4154:
4151:
4133:
4131:
4129:
4128:
4125:
4122:
4100:
4099:
4097:
4096:
4093:
4090:
4075:The Dürer folium
4069:
4052:
4048:
4033:
4018:
4017:
4015:
4014:
4011:
4008:
3993:
3992:
3990:
3989:
3986:
3983:
3969:
3965:
3961:
3957:
3950:
3943:
3939:
3930:
3924:
3920:
3912:
3905:
3901:
3897:
3893:
3892:
3890:
3889:
3883:
3880:
3867:
3865:
3864:
3858:
3855:
3848:
3846:
3845:
3842:
3839:
3831:
3827:
3823:
3822:
3820:
3819:
3814:
3811:
3793:
3785:
3776:
3775:
3773:
3772:
3766:
3763:
3753:
3749:
3745:
3738:
3731:
3729:
3728:
3723:
3721:
3714:
3711:
3707:
3705:
3700:
3699:
3698:
3685:
3680:
3676:
3675:
3673:
3665:
3642:
3637:
3629:
3622:
3617:
3616:
3607:
3591:
3590:
3556:
3551:
3542:
3534:
3525:
3522:
3518:
3516:
3511:
3510:
3509:
3496:
3491:
3487:
3486:
3484:
3476:
3453:
3440:
3435:
3434:
3425:
3409:
3408:
3374:
3366:
3357:
3349:
3333:
3329:
3314:
3284:
3282:
3281:
3276:
3274:
3273:
3268:
3264:
3263:
3262:
3244:
3243:
3234:
3233:
3215:
3214:
3196:
3195:
3180:
3179:
3174:
3170:
3169:
3168:
3156:
3155:
3127:
3125:
3124:
3119:
3117:
3116:
3111:
3107:
3106:
3105:
3093:
3092:
3083:
3082:
3067:
3066:
3057:
3056:
3041:
3040:
3025:
3024:
3012:
3011:
3006:
3002:
3001:
3000:
2988:
2987:
2958:
2947:
2937:The dodecafolium
2929:
2927:
2926:
2921:
2919:
2915:
2914:
2913:
2901:
2900:
2891:
2890:
2872:
2871:
2848:
2847:
2842:
2838:
2837:
2836:
2824:
2823:
2795:
2793:
2792:
2787:
2785:
2781:
2780:
2779:
2761:
2760:
2751:
2750:
2735:
2734:
2714:
2713:
2708:
2704:
2703:
2702:
2690:
2689:
2663:regular pentagon
2660:
2649:
2631:
2629:
2628:
2623:
2621:
2620:
2615:
2611:
2610:
2609:
2594:
2593:
2575:
2574:
2559:
2558:
2553:
2549:
2548:
2547:
2535:
2534:
2506:
2504:
2503:
2498:
2496:
2495:
2490:
2486:
2485:
2484:
2472:
2471:
2462:
2461:
2446:
2445:
2430:
2429:
2417:
2416:
2411:
2407:
2406:
2405:
2393:
2392:
2363:
2352:
2334:
2332:
2331:
2326:
2324:
2320:
2319:
2318:
2303:
2302:
2279:
2278:
2273:
2269:
2268:
2267:
2255:
2254:
2226:
2224:
2223:
2218:
2216:
2212:
2211:
2210:
2192:
2191:
2171:
2170:
2165:
2161:
2160:
2159:
2147:
2146:
2117:
2110:
2092:
2090:
2089:
2084:
2082:
2081:
2076:
2072:
2050:
2049:
2044:
2040:
2039:
2038:
2026:
2025:
1997:
1995:
1994:
1989:
1987:
1986:
1981:
1977:
1976:
1975:
1963:
1962:
1947:
1946:
1934:
1933:
1928:
1924:
1923:
1922:
1910:
1909:
1880:
1869:
1859:The quadrifolium
1851:
1849:
1848:
1843:
1841:
1840:
1835:
1831:
1823:
1813:
1812:
1807:
1803:
1802:
1794:
1777:
1776:
1754:
1752:
1751:
1746:
1744:
1743:
1738:
1734:
1726:
1716:
1715:
1703:
1702:
1697:
1693:
1692:
1684:
1660:
1641:
1625:
1621:
1613:
1609:
1590:
1586:
1585:
1583:
1582:
1577:
1574:
1560:
1550:
1543:
1539:
1529:
1522:
1518:
1500:
1493:
1492:
1490:
1489:
1484:
1481:
1472:
1470:
1469:
1463:
1460:
1439:
1429:
1422:
1415:
1406:
1396:
1389:
1384:
1382:
1381:
1376:
1373:
1362:
1350:
1336:
1332:
1328:
1324:
1317:
1310:
1302:
1292:and will form a
1291:
1284:
1277:
1270:
1251:
1243:
1236:
1228:
1221:
1202:
1177:
1154:
1153:
1151:
1150:
1147:
1144:
1130:
1110:
1088:
1081:
1047:
1040:
1039:
1037:
1036:
1033:
1030:
1023:
1021:
1020:
1017:
1014:
1007:
1005:
1004:
1001:
998:
986:
985:
983:
982:
979:
976:
969:
963:
961:
960:
957:
954:
946:
931:
930:
928:
927:
924:
921:
914:
908:
906:
905:
902:
899:
891:
876:
875:
873:
872:
869:
866:
859:
851:
841:
834:
827:
820:
817:When the period
813:
803:
801:
793:
789:
782:
778:
770:
769:
767:
766:
763:
760:
747:
745:
744:
741:
738:
727:
712:
701:
694:
693:
691:
690:
685:
682:
673:
671:
670:
667:
664:
654:
647:
646:
644:
643:
638:
635:
590:algebraic curves
583:
579:
575:
571:
563:
550:polar coordinate
544:
543:
541:
540:
534:
531:
521:
506:
487:
485:
484:
479:
477:
473:
472:
468:
467:
465:
454:
424:
420:
419:
411:
355:
353:
352:
347:
345:
186:
184:
183:
178:
128:General overview
83:
79:
66:
65:
63:
62:
57:
54:
40:
5343:
5342:
5338:
5337:
5336:
5334:
5333:
5332:
5318:
5317:
5293:
5288:
5283:
5282:
5273:
5271:
5261:
5257:
5248:
5246:
5240:
5236:
5227:
5225:
5221:Jan Wassenaar.
5219:
5215:
5206:
5204:
5198:
5194:
5185:
5183:
5177:
5173:
5164:
5162:
5156:
5152:
5143:
5141:
5137:Jan Wassenaar.
5135:
5131:
5122:
5120:
5114:
5110:
5101:
5099:
5089:
5085:
5076:
5074:
5070:
5069:
5065:
5056:
5054:
5048:
5044:
5035:
5033:
5025:
5024:
5020:
5011:
5009:
4999:
4995:
4986:
4984:
4978:
4974:
4965:
4963:
4957:
4953:
4944:
4942:
4938:
4937:
4933:
4927:H. Martyn Cundy
4919:
4915:
4891:
4887:
4882:
4864:Rose (topology)
4849:
4835:
4832:
4829:
4828:
4826:
4821:
4814:
4799:
4787:
4779:
4776:
4773:
4772:
4770:
4759:
4756:
4753:
4752:
4750:
4748:
4741:
4726:
4722:
4715:
4710:
4705:
4704:
4703:
4702:
4701:
4691:
4684:
4677:
4669:
4666:
4663:
4662:
4660:
4655:
4652:
4644:
4643:
4633:
4626:
4619:
4612:
4611:The trifolium,
4609:
4601:
4600:
4593:
4590:
4587:
4586:
4584:
4574:
4567:
4560:
4552:
4549:
4546:
4545:
4543:
4538:
4531:
4523:
4522:
4512:
4505:
4498:
4491:
4484:
4475:
4474:
4467:
4457:
4456:is an integer,
4453:
4443:
4441:
4433:
4430:
4425:
4424:
4422:
4417:
4415:
4404:
4382:
4379:
4376:
4375:
4373:
4368:
4365:
4336:
4332:
4326:
4322:
4313:
4302:
4298:
4284:
4280:
4271:
4267:
4266:
4262:
4258:
4254:
4253:
4242:
4238:
4229:
4225:
4224:
4220:
4218:
4215:
4214:
4202:
4199:
4194:
4193:
4191:
4180:
4177:
4172:
4171:
4169:
4164:
4155:
4152:
4147:
4146:
4144:
4135:
4126:
4123:
4118:
4117:
4115:
4106:
4094:
4091:
4088:
4087:
4085:
4080:
4077:
4061:
4050:
4035:
4020:
4012:
4009:
4003:
4002:
4000:
3995:
3987:
3984:
3979:
3978:
3976:
3971:
3967:
3963:
3959:
3955:
3945:
3941:
3937:
3926:
3922:
3918:
3907:
3903:
3899:
3895:
3884:
3881:
3872:
3871:
3869:
3859:
3856:
3853:
3852:
3850:
3843:
3840:
3837:
3836:
3834:
3833:
3829:
3825:
3815:
3812:
3807:
3806:
3804:
3799:
3791:
3788:
3783:
3767:
3764:
3759:
3758:
3756:
3755:
3751:
3747:
3740:
3736:
3719:
3718:
3710:
3706:
3694:
3690:
3686:
3684:
3666:
3643:
3641:
3628:
3627:
3623:
3612:
3608:
3606:
3599:
3586:
3582:
3552:
3547:
3533:
3530:
3529:
3521:
3517:
3505:
3501:
3497:
3495:
3477:
3454:
3452:
3445:
3441:
3430:
3426:
3424:
3417:
3404:
3400:
3367:
3362:
3348:
3344:
3342:
3339:
3338:
3331:
3316:
3301:
3294:
3269:
3258:
3254:
3239:
3235:
3229:
3225:
3210:
3206:
3202:
3198:
3197:
3191:
3187:
3175:
3164:
3160:
3151:
3147:
3146:
3142:
3141:
3139:
3136:
3135:
3112:
3101:
3097:
3088:
3084:
3078:
3074:
3062:
3058:
3052:
3048:
3036:
3032:
3031:
3027:
3026:
3020:
3016:
3007:
2996:
2992:
2983:
2979:
2978:
2974:
2973:
2971:
2968:
2967:
2953:
2952:because it has
2942:
2939:
2909:
2905:
2896:
2892:
2886:
2882:
2867:
2863:
2859:
2855:
2843:
2832:
2828:
2819:
2815:
2814:
2810:
2809:
2807:
2804:
2803:
2775:
2771:
2756:
2752:
2746:
2742:
2730:
2726:
2725:
2721:
2709:
2698:
2694:
2685:
2681:
2680:
2676:
2675:
2673:
2670:
2669:
2655:
2654:because it has
2644:
2641:
2639:The pentafolium
2616:
2605:
2601:
2589:
2585:
2581:
2577:
2576:
2570:
2566:
2554:
2543:
2539:
2530:
2526:
2525:
2521:
2520:
2518:
2515:
2514:
2491:
2480:
2476:
2467:
2463:
2457:
2453:
2441:
2437:
2436:
2432:
2431:
2425:
2421:
2412:
2401:
2397:
2388:
2384:
2383:
2379:
2378:
2376:
2373:
2372:
2358:
2357:because it has
2347:
2344:
2314:
2310:
2298:
2294:
2290:
2286:
2274:
2263:
2259:
2250:
2246:
2245:
2241:
2240:
2238:
2235:
2234:
2206:
2202:
2187:
2183:
2182:
2178:
2166:
2155:
2151:
2142:
2138:
2137:
2133:
2132:
2130:
2127:
2126:
2112:
2105:
2102:
2077:
2062:
2058:
2057:
2045:
2034:
2030:
2021:
2017:
2016:
2012:
2011:
2009:
2006:
2005:
1982:
1971:
1967:
1958:
1954:
1953:
1949:
1948:
1942:
1938:
1929:
1918:
1914:
1905:
1901:
1900:
1896:
1895:
1893:
1890:
1889:
1875:
1874:because it has
1864:
1861:
1836:
1822:
1818:
1817:
1808:
1793:
1786:
1782:
1781:
1772:
1768:
1766:
1763:
1762:
1739:
1725:
1721:
1720:
1711:
1707:
1698:
1683:
1676:
1672:
1671:
1669:
1666:
1665:
1647:
1636:
1633:
1623:
1615:
1611:
1604:
1588:
1578:
1575:
1569:
1568:
1566:
1565:
1552:
1545:
1541:
1531:
1524:
1520:
1516:
1495:
1485:
1482:
1477:
1476:
1474:
1464:
1461:
1455:
1454:
1452:
1451:
1431:
1424:
1417:
1413:
1398:
1391:
1377:
1374:
1368:
1367:
1365:
1364:
1357:
1342:
1334:
1330:
1326:
1322:
1312:
1308:
1297:
1286:
1279:
1272:
1261:
1249:
1238:
1230:
1223:
1212:
1205:
1200:
1156:
1148:
1145:
1140:
1139:
1137:
1132:
1117:
1093:
1083:
1068:
1057:
1045:
1034:
1031:
1028:
1027:
1025:
1018:
1015:
1012:
1011:
1009:
1002:
999:
996:
995:
993:
988:
980:
977:
974:
973:
971:
965:
958:
955:
952:
951:
949:
948:
933:
925:
922:
919:
918:
916:
910:
903:
900:
897:
896:
894:
893:
878:
870:
867:
864:
863:
861:
855:
853:
843:
836:
829:
822:
818:
805:
797:
795:
791:
787:
780:
776:
764:
761:
756:
755:
753:
742:
739:
734:
733:
731:
729:
714:
706:
696:
686:
683:
678:
677:
675:
668:
665:
660:
659:
657:
656:
649:
639:
636:
630:
629:
627:
622:
614:
598:
586:rational number
581:
577:
573:
569:
561:
535:
532:
527:
526:
524:
523:
508:
493:
458:
453:
446:
442:
438:
434:
410:
400:
396:
367:
364:
363:
343:
342:
278:
272:
271:
207:
200:
198:
195:
194:
148:
145:
144:
135:
130:
81:
70:
68:
58:
55:
50:
49:
47:
42:
31:
24:
21:Rose (topology)
17:
12:
11:
5:
5341:
5331:
5330:
5316:
5315:
5309:
5304:
5298:
5287:
5286:External links
5284:
5281:
5280:
5255:
5244:"Dürer Folium"
5234:
5213:
5192:
5171:
5150:
5129:
5108:
5083:
5063:
5042:
5018:
4993:
4972:
4951:
4931:
4913:
4884:
4883:
4881:
4878:
4877:
4876:
4871:
4866:
4861:
4856:
4843:
4813:
4810:
4714:
4706:
4653:
4646:
4645:
4610:
4603:
4602:
4532:
4525:
4524:
4485:
4478:
4477:
4476:
4402:
4401:
4400:
4399:
4364:
4361:
4353:
4352:
4339:
4335:
4329:
4325:
4321:
4316:
4311:
4305:
4301:
4297:
4293:
4287:
4283:
4279:
4274:
4270:
4265:
4261:
4257:
4251:
4245:
4241:
4237:
4232:
4228:
4223:
4103:Albrecht Dürer
4076:
4073:
4072:
4071:
4057:
4056:
4055:
4054:
3933:
3932:
3787:
3779:
3733:
3732:
3717:
3708:
3703:
3697:
3693:
3689:
3683:
3679:
3672:
3669:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3640:
3635:
3632:
3626:
3620:
3615:
3611:
3605:
3602:
3600:
3598:
3595:
3589:
3585:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3555:
3550:
3546:
3540:
3537:
3532:
3531:
3528:
3523:for even
3519:
3514:
3508:
3504:
3500:
3494:
3490:
3483:
3480:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3451:
3448:
3444:
3438:
3433:
3429:
3423:
3420:
3418:
3416:
3413:
3407:
3403:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3373:
3370:
3365:
3361:
3355:
3352:
3347:
3346:
3293:
3290:
3288:respectively.
3286:
3285:
3272:
3267:
3261:
3257:
3253:
3250:
3247:
3242:
3238:
3232:
3228:
3224:
3221:
3218:
3213:
3209:
3205:
3201:
3194:
3190:
3186:
3183:
3178:
3173:
3167:
3163:
3159:
3154:
3150:
3145:
3129:
3128:
3115:
3110:
3104:
3100:
3096:
3091:
3087:
3081:
3077:
3073:
3070:
3065:
3061:
3055:
3051:
3047:
3044:
3039:
3035:
3030:
3023:
3019:
3015:
3010:
3005:
2999:
2995:
2991:
2986:
2982:
2977:
2938:
2935:
2933:respectively.
2931:
2930:
2918:
2912:
2908:
2904:
2899:
2895:
2889:
2885:
2881:
2878:
2875:
2870:
2866:
2862:
2858:
2854:
2851:
2846:
2841:
2835:
2831:
2827:
2822:
2818:
2813:
2797:
2796:
2784:
2778:
2774:
2770:
2767:
2764:
2759:
2755:
2749:
2745:
2741:
2738:
2733:
2729:
2724:
2720:
2717:
2712:
2707:
2701:
2697:
2693:
2688:
2684:
2679:
2640:
2637:
2635:respectively.
2633:
2632:
2619:
2614:
2608:
2604:
2600:
2597:
2592:
2588:
2584:
2580:
2573:
2569:
2565:
2562:
2557:
2552:
2546:
2542:
2538:
2533:
2529:
2524:
2508:
2507:
2494:
2489:
2483:
2479:
2475:
2470:
2466:
2460:
2456:
2452:
2449:
2444:
2440:
2435:
2428:
2424:
2420:
2415:
2410:
2404:
2400:
2396:
2391:
2387:
2382:
2343:
2342:The octafolium
2340:
2336:
2335:
2323:
2317:
2313:
2309:
2306:
2301:
2297:
2293:
2289:
2285:
2282:
2277:
2272:
2266:
2262:
2258:
2253:
2249:
2244:
2228:
2227:
2215:
2209:
2205:
2201:
2198:
2195:
2190:
2186:
2181:
2177:
2174:
2169:
2164:
2158:
2154:
2150:
2145:
2141:
2136:
2101:
2098:
2096:respectively.
2094:
2093:
2080:
2075:
2071:
2068:
2065:
2061:
2056:
2053:
2048:
2043:
2037:
2033:
2029:
2024:
2020:
2015:
1999:
1998:
1985:
1980:
1974:
1970:
1966:
1961:
1957:
1952:
1945:
1941:
1937:
1932:
1927:
1921:
1917:
1913:
1908:
1904:
1899:
1860:
1857:
1855:respectively.
1853:
1852:
1839:
1834:
1829:
1826:
1821:
1816:
1811:
1806:
1800:
1797:
1792:
1789:
1785:
1780:
1775:
1771:
1756:
1755:
1742:
1737:
1732:
1729:
1724:
1719:
1714:
1710:
1706:
1701:
1696:
1690:
1687:
1682:
1679:
1675:
1632:
1629:
1628:
1627:
1600:
1599:
1595:
1594:
1593:
1592:
1512:
1511:
1510:
1509:
1502:
1448:
1409:
1408:
1353:
1352:
1204:
1196:
1195:
1194:
1185:
1184:
1180:
1179:
1113:
1112:
1056:
1053:
1052:
1051:
1050:
1049:
1042:
815:
784:
613:
610:
597:
594:
490:
489:
476:
471:
464:
461:
457:
452:
449:
445:
441:
437:
433:
430:
427:
423:
417:
414:
409:
406:
403:
399:
395:
392:
389:
386:
383:
380:
377:
374:
371:
357:
356:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
279:
277:
274:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
208:
206:
203:
202:
188:
187:
176:
173:
170:
167:
164:
161:
158:
155:
152:
139:polar equation
134:
131:
129:
126:
98:rhodonea curve
15:
9:
6:
4:
3:
2:
5340:
5329:
5326:
5325:
5323:
5313:
5310:
5308:
5305:
5302:
5299:
5297:
5290:
5289:
5270:
5266:
5259:
5245:
5238:
5224:
5217:
5203:
5196:
5182:
5175:
5161:
5154:
5140:
5133:
5119:
5112:
5098:
5094:
5087:
5073:
5067:
5053:
5046:
5032:
5031:ProofWiki.org
5028:
5022:
5008:
5004:
4997:
4983:
4976:
4962:
4955:
4941:
4935:
4928:
4924:
4923:
4917:
4910:
4906:
4905:
4900:
4896:
4889:
4885:
4875:
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4852:
4847:
4844:
4824:
4819:
4816:
4815:
4809:
4806:
4802:
4797:
4791:
4768:
4744:
4740:has a period
4737:
4733:
4729:
4720:
4713:
4698:
4694:
4687:
4680:
4658:
4650:
4640:
4636:
4629:
4622:
4615:
4607:
4581:
4577:
4570:
4563:
4541:
4536:
4529:
4519:
4515:
4508:
4501:
4494:
4489:
4482:
4470:
4464:
4460:
4450:
4446:
4428:
4420:
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4407:
4398:
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4392:
4371:
4360:
4358:
4337:
4333:
4327:
4323:
4319:
4314:
4309:
4303:
4299:
4295:
4291:
4285:
4281:
4277:
4272:
4268:
4263:
4259:
4255:
4249:
4243:
4239:
4235:
4230:
4226:
4221:
4213:
4212:
4211:
4197:
4189:
4175:
4167:
4150:
4142:
4138:
4121:
4113:
4109:
4104:
4083:
4068:
4064:
4059:
4058:
4046:
4042:
4038:
4031:
4027:
4023:
4007:
3998:
3982:
3974:
3953:
3952:
3949:
3935:
3934:
3929:
3916:
3915:
3914:
3911:
3906:are odd, and
3888:
3879:
3875:
3863:
3818:
3810:
3802:
3797:
3786:
3778:
3771:
3762:
3744:
3715:
3712:for odd
3701:
3695:
3691:
3687:
3681:
3677:
3670:
3667:
3659:
3656:
3653:
3647:
3644:
3638:
3633:
3630:
3624:
3618:
3613:
3609:
3603:
3601:
3596:
3593:
3587:
3576:
3573:
3567:
3564:
3561:
3553:
3548:
3544:
3538:
3535:
3526:
3512:
3506:
3502:
3498:
3492:
3488:
3481:
3478:
3470:
3467:
3464:
3458:
3455:
3449:
3446:
3442:
3436:
3431:
3427:
3421:
3419:
3414:
3411:
3405:
3394:
3391:
3385:
3382:
3379:
3371:
3368:
3363:
3359:
3353:
3350:
3337:
3336:
3335:
3327:
3323:
3319:
3312:
3308:
3304:
3299:
3289:
3270:
3265:
3259:
3255:
3251:
3248:
3245:
3240:
3236:
3230:
3226:
3222:
3219:
3216:
3211:
3207:
3203:
3199:
3192:
3188:
3184:
3181:
3176:
3171:
3165:
3161:
3157:
3152:
3148:
3143:
3134:
3133:
3132:
3113:
3108:
3102:
3098:
3094:
3089:
3085:
3079:
3075:
3071:
3068:
3063:
3059:
3053:
3049:
3045:
3042:
3037:
3033:
3028:
3021:
3017:
3013:
3008:
3003:
2997:
2993:
2989:
2984:
2980:
2975:
2966:
2965:
2964:
2962:
2956:
2951:
2945:
2934:
2916:
2910:
2906:
2902:
2897:
2893:
2887:
2883:
2879:
2876:
2873:
2868:
2864:
2860:
2856:
2852:
2849:
2844:
2839:
2833:
2829:
2825:
2820:
2816:
2811:
2802:
2801:
2800:
2782:
2776:
2772:
2768:
2765:
2762:
2757:
2753:
2747:
2743:
2739:
2736:
2731:
2727:
2722:
2718:
2715:
2710:
2705:
2699:
2695:
2691:
2686:
2682:
2677:
2668:
2667:
2666:
2664:
2658:
2653:
2647:
2636:
2617:
2612:
2606:
2602:
2598:
2595:
2590:
2586:
2582:
2578:
2571:
2567:
2563:
2560:
2555:
2550:
2544:
2540:
2536:
2531:
2527:
2522:
2513:
2512:
2511:
2492:
2487:
2481:
2477:
2473:
2468:
2464:
2458:
2454:
2450:
2447:
2442:
2438:
2433:
2426:
2422:
2418:
2413:
2408:
2402:
2398:
2394:
2389:
2385:
2380:
2371:
2370:
2369:
2367:
2361:
2356:
2353:is called an
2350:
2339:
2321:
2315:
2311:
2307:
2304:
2299:
2295:
2291:
2287:
2283:
2280:
2275:
2270:
2264:
2260:
2256:
2251:
2247:
2242:
2233:
2232:
2231:
2213:
2207:
2203:
2199:
2196:
2193:
2188:
2184:
2179:
2175:
2172:
2167:
2162:
2156:
2152:
2148:
2143:
2139:
2134:
2125:
2124:
2123:
2121:
2115:
2108:
2100:The trifolium
2097:
2078:
2073:
2069:
2066:
2063:
2059:
2054:
2051:
2046:
2041:
2035:
2031:
2027:
2022:
2018:
2013:
2004:
2003:
2002:
1983:
1978:
1972:
1968:
1964:
1959:
1955:
1950:
1943:
1939:
1935:
1930:
1925:
1919:
1915:
1911:
1906:
1902:
1897:
1888:
1887:
1886:
1884:
1878:
1873:
1867:
1856:
1837:
1832:
1827:
1824:
1819:
1814:
1809:
1804:
1798:
1795:
1790:
1787:
1783:
1778:
1773:
1769:
1761:
1760:
1759:
1740:
1735:
1730:
1727:
1722:
1717:
1712:
1708:
1704:
1699:
1694:
1688:
1685:
1680:
1677:
1673:
1664:
1663:
1662:
1658:
1654:
1650:
1645:
1639:
1619:
1607:
1602:
1601:
1597:
1596:
1581:
1573:
1563:
1562:
1559:
1555:
1548:
1538:
1534:
1528:
1514:
1513:
1507:
1503:
1499:
1488:
1480:
1468:
1459:
1449:
1446:
1445:
1443:
1438:
1434:
1428:
1421:
1411:
1410:
1405:
1401:
1394:
1388:
1380:
1372:
1361:
1355:
1354:
1349:
1345:
1340:
1339:
1338:
1325:is even, and
1320:
1316:
1300:
1295:
1289:
1282:
1275:
1268:
1264:
1258:
1247:
1241:
1234:
1226:
1219:
1215:
1209:
1203:
1192:
1187:
1186:
1182:
1181:
1175:
1171:
1167:
1163:
1159:
1143:
1135:
1128:
1124:
1120:
1115:
1114:
1108:
1104:
1100:
1096:
1092:
1086:
1079:
1075:
1071:
1066:
1065:
1064:
1062:
1043:
991:
968:
945:
941:
937:
913:
890:
886:
882:
858:
850:
846:
840:
833:
826:
816:
812:
808:
800:
785:
774:
759:
751:
737:
725:
721:
717:
713:specified by
710:
704:
703:
699:
689:
681:
663:
652:
642:
634:
625:
620:
616:
615:
609:
602:
593:
591:
587:
580:(though when
567:
559:
555:
552:(rather than
551:
546:
539:
530:
519:
515:
511:
504:
500:
496:
474:
469:
462:
459:
455:
450:
447:
443:
439:
435:
431:
428:
425:
421:
415:
412:
407:
404:
401:
397:
393:
390:
387:
381:
378:
372:
369:
362:
361:
360:
336:
330:
327:
321:
318:
312:
309:
306:
303:
297:
291:
288:
285:
282:
280:
275:
265:
259:
256:
250:
247:
241:
238:
235:
232:
226:
220:
217:
214:
211:
209:
204:
193:
192:
191:
171:
168:
162:
159:
156:
153:
150:
143:
142:
141:
140:
133:Specification
125:
123:
119:
115:
111:
107:
103:
99:
95:
91:
77:
73:
61:
53:
45:
38:
34:
28:
22:
5328:Plane curves
5272:. Retrieved
5268:
5258:
5247:. Retrieved
5237:
5226:. Retrieved
5216:
5205:. Retrieved
5202:"Rose Curve"
5195:
5184:. Retrieved
5181:"Rose Curve"
5174:
5163:. Retrieved
5153:
5142:. Retrieved
5132:
5121:. Retrieved
5111:
5100:. Retrieved
5096:
5086:
5075:. Retrieved
5066:
5055:. Retrieved
5045:
5034:. Retrieved
5030:
5021:
5010:. Retrieved
5006:
4996:
4985:. Retrieved
4982:"Rose Curve"
4975:
4964:. Retrieved
4954:
4943:. Retrieved
4934:
4920:
4916:
4902:
4888:
4850:
4846:Quadrifolium
4822:
4804:
4800:
4789:
4766:
4742:
4735:
4731:
4727:
4716:
4709:
4696:
4692:
4685:
4678:
4656:
4638:
4634:
4627:
4620:
4613:
4583:is reached (
4579:
4575:
4568:
4561:
4539:
4517:
4513:
4506:
4499:
4492:
4468:
4462:
4458:
4448:
4444:
4426:
4418:
4409:
4405:
4369:
4367:A rose with
4366:
4354:
4195:
4187:
4173:
4165:
4148:
4140:
4136:
4119:
4111:
4107:
4081:
4079:A rose with
4078:
4066:
4062:
4044:
4040:
4036:
4029:
4025:
4021:
4005:
3996:
3980:
3972:
3947:
3940:is even and
3927:
3909:
3886:
3877:
3873:
3861:
3816:
3808:
3800:
3789:
3782:
3769:
3760:
3742:
3734:
3325:
3321:
3317:
3310:
3306:
3302:
3295:
3287:
3130:
2954:
2950:dodecafolium
2948:is called a
2943:
2941:A rose with
2940:
2932:
2798:
2656:
2650:is called a
2645:
2643:A rose with
2642:
2634:
2509:
2359:
2348:
2346:A rose with
2345:
2337:
2229:
2113:
2106:
2104:A rose with
2103:
2095:
2000:
1876:
1872:quadrifolium
1870:is called a
1865:
1863:A rose with
1862:
1854:
1757:
1656:
1652:
1648:
1637:
1635:A rose with
1634:
1617:
1614:is odd, and
1605:
1579:
1571:
1557:
1553:
1546:
1536:
1532:
1526:
1497:
1486:
1478:
1466:
1457:
1436:
1432:
1426:
1419:
1403:
1399:
1392:
1386:
1378:
1370:
1363:, there are
1359:
1347:
1343:
1329:petals when
1318:
1314:
1306:
1298:
1287:
1280:
1273:
1266:
1262:
1239:
1232:
1224:
1217:
1213:
1199:
1173:
1169:
1165:
1161:
1157:
1141:
1133:
1126:
1122:
1118:
1106:
1102:
1098:
1094:
1084:
1077:
1073:
1069:
1058:
989:
970:| <
966:
943:
939:
935:
915:| <
911:
888:
884:
880:
860:| <
856:
848:
844:
838:
831:
824:
810:
806:
798:
757:
749:
735:
723:
719:
715:
708:
697:
687:
679:
661:
650:
640:
632:
623:
618:
607:
547:
537:
528:
517:
513:
509:
502:
498:
494:
491:
358:
189:
136:
122:Guido Grandi
97:
93:
87:
75:
71:
59:
51:
43:
36:
32:
5072:"Trifolium"
4859:Maurer rose
3958:is odd and
3913:otherwise.
2652:pentafolium
114:phase angle
90:mathematics
5274:2021-02-05
5249:2021-02-03
5228:2021-02-02
5223:"Rhodonea"
5207:2021-02-12
5186:2021-02-12
5165:2021-02-05
5144:2021-02-02
5139:"Rhodonea"
5123:2021-02-03
5102:2021-02-05
5077:2021-02-02
5057:2021-02-03
5036:2021-02-03
5012:2021-02-05
4987:2021-02-12
4966:2021-02-03
4945:2021-02-02
4899:"Rhodonea"
4874:Spirograph
4395:trisectrix
4357:trisectrix
3296:The total
2355:octafolium
1631:The circle
5296:parameter
5200:Xah Lee.
5179:Xah Lee.
4980:Xah Lee.
4796:dense set
4296:−
3688:π
3660:π
3648:
3631:π
3597:θ
3577:θ
3568:
3554:π
3545:∫
3499:π
3471:π
3459:
3447:π
3415:θ
3395:θ
3386:
3372:π
3360:∫
3220:−
3095:−
3043:−
2961:dodecagon
2877:−
2737:−
2596:−
2448:−
2308:−
2194:−
1965:−
1791:−
1681:−
1271:. Since
1211:The rose
773:rotations
566:amplitude
456:π
451:−
448:θ
432:
413:π
408:−
405:θ
394:
382:θ
373:
337:θ
331:
322:θ
313:
298:θ
292:
266:θ
260:
251:θ
242:
227:θ
221:
172:θ
163:
5322:Category
4812:See also
3898:if both
3824:, where
3330:, where
1626:is even.
1294:heptagon
1222:. Since
1091:identity
1061:symmetry
1055:Symmetry
964:≤ |
909:≤ |
102:sinusoid
5303:Xah Lee
4839:
4827:
4783:
4771:
4763:
4751:
4673:
4661:
4597:
4585:
4556:
4544:
4437:
4423:
4386:
4374:
4206:
4192:
4184:
4170:
4159:
4145:
4130:
4116:
4098:
4086:
4016:
4001:
3991:
3977:
3891:
3870:
3866:
3851:
3847:
3835:
3821:
3805:
3774:
3757:
2366:octagon
1584:
1567:
1506:apothem
1491:
1475:
1471:
1453:
1442:polygon
1383:
1366:
1265:= cos(7
1246:octagon
1216:= cos(4
1152:
1138:
1038:
1026:
1022:
1010:
1006:
994:
984:
972:
962:
950:
929:
917:
907:
895:
874:
862:
842:. When
768:
754:
746:
732:
692:
676:
672:
658:
655:and is
645:
628:
564:and an
542:
525:
64:
48:
5160:"Rose"
5118:"Rose"
5052:"Rose"
4961:"Rose"
4488:circle
4408:= cos(
3994:or at
1883:square
1644:circle
1319:petals
1307:When
854:|
847:> 4
802:|
796:|
619:petals
612:Petals
106:cosine
74:= sin(
35:= cos(
4880:Notes
4452:when
4389:is a
3936:When
3931:long.
3798:form
3735:When
1642:is a
1622:when
1610:when
1515:When
1412:When
1250:(1,0)
1105:cos(−
938:<
883:<
584:is a
100:is a
4734:cos(
4721:for
4695:= 10
4533:The
4486:The
4190:sin(
4186:) ≠
4168:cos(
4143:sin(
4134:and
4114:cos(
4043:sin(
4034:and
4028:cos(
3966:and
3921:and
3902:and
3828:and
3324:sin(
3309:cos(
3298:area
3131:and
2957:= 12
2799:and
2510:and
2230:and
2001:and
1758:and
1655:cos(
1620:+ 1)
1168:sin(
1164:) =
1160:sin(
1125:sin(
1101:) =
1097:cos(
1076:cos(
947:(or
942:≤ 12
892:(or
852:(or
779:and
722:cos(
516:cos(
501:sin(
110:sine
94:rose
92:, a
4925:by
4853:= 2
4808:).
4792:,0)
4745:= 2
4688:= 5
4681:= 4
4630:= 1
4623:= 3
4616:= 3
4578:= 3
4571:= 3
4564:= 1
4509:= 1
4502:= 1
4495:= 1
4471:= 0
4461:= 2
4447:= 2
3645:sin
3565:cos
3456:sin
3383:cos
3315:or
2946:= 6
2659:= 5
2648:= 5
2362:= 8
2351:= 4
2116:= 3
2109:= 3
1879:= 4
1868:= 2
1640:= 1
1608:+ 1
1551:to
1549:= 0
1402:= 2
1395:= 0
1321:if
1301:= 1
1290:= 1
1283:= 7
1276:= 7
1242:= 1
1235:= 8
1227:= 4
1087:= 0
887:≤ 8
711:,0)
702:.)
700:≤ 0
653:≥ 0
592:).
568:of
560:of
429:cos
391:cos
370:sin
328:sin
310:cos
289:sin
257:cos
239:cos
218:cos
160:cos
108:or
96:or
88:In
5324::
5267:.
5095:.
5029:.
5005:.
4907:,
4901:,
4897:,
4825:=
4803:≤
4769:≤
4765:≤
4736:πθ
4730:=
4683:,
4659:=
4637:=
4625:,
4566:,
4542:=
4537:,
4516:=
4504:,
4490:,
4463:dπ
4421:=
4410:kθ
4372:=
4139:=
4110:=
4084:=
4065:=
4045:kθ
4039:=
4030:kθ
4024:=
4006:dπ
3999:=
3981:dπ
3975:=
3948:dπ
3928:dπ
3876:−
3868:=
3849:−
3803:=
3777:.
3761:πa
3326:kθ
3320:=
3311:kθ
3305:=
3223:10
3072:15
3046:15
2955:2k
2880:10
2740:10
2564:16
2360:2k
1877:2k
1651:=
1616:2(
1556:=
1535:=
1473:=
1435:=
1385:=
1346:=
1174:kθ
1172:−
1162:kθ
1136:=
1127:kθ
1121:=
1107:kθ
1099:kθ
1078:kθ
1072:=
1024:,
1008:,
992:=
809:=
752:≤
748:≤
724:kθ
718:=
674:=
626:=
518:kθ
512:=
503:kθ
497:=
76:kθ
67:.
46:=
37:kθ
5294:k
5277:.
5252:.
5231:.
5210:.
5189:.
5168:.
5147:.
5126:.
5105:.
5080:.
5060:.
5039:.
5015:.
4990:.
4969:.
4948:.
4855:.
4851:k
4842:.
4836:3
4833:/
4830:1
4823:k
4805:a
4801:r
4790:a
4788:(
4780:2
4777:/
4774:1
4767:θ
4760:2
4757:/
4754:1
4749:−
4743:T
4738:)
4732:a
4728:r
4723:k
4711:k
4697:π
4693:θ
4686:d
4679:n
4676:(
4670:5
4667:/
4664:4
4657:k
4639:π
4635:θ
4628:d
4621:n
4618:(
4614:k
4594:2
4591:/
4588:3
4580:π
4576:θ
4569:d
4562:n
4559:(
4553:3
4550:/
4547:1
4540:k
4518:π
4514:θ
4507:d
4500:n
4497:(
4493:k
4473:.
4469:θ
4459:θ
4454:k
4449:π
4445:θ
4440:.
4434:2
4431:/
4427:π
4419:θ
4412:)
4406:r
4383:3
4380:/
4377:1
4370:k
4338:2
4334:x
4328:4
4324:a
4320:=
4315:2
4310:)
4304:2
4300:a
4292:)
4286:2
4282:y
4278:+
4273:2
4269:x
4264:(
4260:2
4256:(
4250:)
4244:2
4240:y
4236:+
4231:2
4227:x
4222:(
4208:)
4203:2
4200:/
4196:θ
4188:a
4181:2
4178:/
4174:θ
4166:a
4161:)
4156:2
4153:/
4149:θ
4141:a
4137:r
4132:)
4127:2
4124:/
4120:θ
4112:a
4108:r
4095:2
4092:/
4089:1
4082:k
4067:a
4063:r
4051:k
4047:)
4041:a
4037:r
4032:)
4026:a
4022:r
4013:2
4010:/
4004:3
3997:θ
3988:2
3985:/
3973:θ
3968:k
3964:a
3960:d
3956:n
3946:2
3942:d
3938:n
3923:d
3919:n
3910:n
3908:2
3904:d
3900:n
3896:n
3887:n
3885:2
3882:/
3878:d
3874:n
3862:k
3860:2
3857:/
3854:1
3844:2
3841:/
3838:1
3830:d
3826:n
3817:d
3813:/
3809:n
3801:k
3792:k
3784:k
3770:k
3768:4
3765:/
3752:k
3748:k
3743:k
3741:2
3737:k
3716:k
3702:4
3696:2
3692:a
3682:=
3678:)
3671:k
3668:4
3663:)
3657:k
3654:2
3651:(
3639:+
3634:2
3625:(
3619:2
3614:2
3610:a
3604:=
3594:d
3588:2
3584:)
3580:)
3574:k
3571:(
3562:a
3559:(
3549:0
3539:2
3536:1
3527:k
3513:2
3507:2
3503:a
3493:=
3489:)
3482:k
3479:4
3474:)
3468:k
3465:4
3462:(
3450:+
3443:(
3437:2
3432:2
3428:a
3422:=
3412:d
3406:2
3402:)
3398:)
3392:k
3389:(
3380:a
3377:(
3369:2
3364:0
3354:2
3351:1
3332:k
3328:)
3322:a
3318:r
3313:)
3307:a
3303:r
3271:2
3266:)
3260:5
3256:y
3252:x
3249:3
3246:+
3241:3
3237:y
3231:3
3227:x
3217:y
3212:5
3208:x
3204:3
3200:(
3193:2
3189:a
3185:4
3182:=
3177:7
3172:)
3166:2
3162:y
3158:+
3153:2
3149:x
3144:(
3114:2
3109:)
3103:6
3099:y
3090:4
3086:y
3080:2
3076:x
3069:+
3064:2
3060:y
3054:4
3050:x
3038:6
3034:x
3029:(
3022:2
3018:a
3014:=
3009:7
3004:)
2998:2
2994:y
2990:+
2985:2
2981:x
2976:(
2944:k
2917:)
2911:5
2907:y
2903:+
2898:3
2894:y
2888:2
2884:x
2874:y
2869:4
2865:x
2861:5
2857:(
2853:a
2850:=
2845:3
2840:)
2834:2
2830:y
2826:+
2821:2
2817:x
2812:(
2783:)
2777:4
2773:y
2769:x
2766:5
2763:+
2758:2
2754:y
2748:3
2744:x
2732:5
2728:x
2723:(
2719:a
2716:=
2711:3
2706:)
2700:2
2696:y
2692:+
2687:2
2683:x
2678:(
2657:k
2646:k
2618:2
2613:)
2607:3
2603:x
2599:y
2591:3
2587:y
2583:x
2579:(
2572:2
2568:a
2561:=
2556:5
2551:)
2545:2
2541:y
2537:+
2532:2
2528:x
2523:(
2493:2
2488:)
2482:4
2478:y
2474:+
2469:2
2465:y
2459:2
2455:x
2451:6
2443:4
2439:x
2434:(
2427:2
2423:a
2419:=
2414:5
2409:)
2403:2
2399:y
2395:+
2390:2
2386:x
2381:(
2349:k
2322:)
2316:3
2312:y
2305:y
2300:2
2296:x
2292:3
2288:(
2284:a
2281:=
2276:2
2271:)
2265:2
2261:y
2257:+
2252:2
2248:x
2243:(
2214:)
2208:2
2204:y
2200:x
2197:3
2189:3
2185:x
2180:(
2176:a
2173:=
2168:2
2163:)
2157:2
2153:y
2149:+
2144:2
2140:x
2135:(
2114:k
2107:k
2079:2
2074:)
2070:y
2067:x
2064:a
2060:(
2055:4
2052:=
2047:3
2042:)
2036:2
2032:y
2028:+
2023:2
2019:x
2014:(
1984:2
1979:)
1973:2
1969:y
1960:2
1956:x
1951:(
1944:2
1940:a
1936:=
1931:3
1926:)
1920:2
1916:y
1912:+
1907:2
1903:x
1898:(
1866:k
1838:2
1833:)
1828:2
1825:a
1820:(
1815:=
1810:2
1805:)
1799:2
1796:a
1788:y
1784:(
1779:+
1774:2
1770:x
1741:2
1736:)
1731:2
1728:a
1723:(
1718:=
1713:2
1709:y
1705:+
1700:2
1695:)
1689:2
1686:a
1678:x
1674:(
1659:)
1657:θ
1653:a
1649:r
1638:k
1624:k
1618:k
1612:k
1606:k
1591:.
1589:k
1580:k
1576:/
1572:π
1570:2
1558:π
1554:θ
1547:θ
1542:π
1537:a
1533:r
1527:π
1525:2
1521:k
1517:k
1501:.
1498:k
1496:2
1487:k
1483:/
1479:π
1467:k
1465:2
1462:/
1458:π
1456:2
1437:a
1433:r
1427:π
1425:2
1420:k
1418:2
1414:k
1404:π
1400:θ
1393:θ
1387:k
1379:T
1375:/
1371:π
1369:2
1360:π
1358:2
1348:a
1344:r
1335:k
1331:k
1327:k
1323:k
1315:k
1313:2
1309:k
1303:.
1299:r
1288:r
1281:k
1274:k
1269:)
1267:θ
1263:r
1240:r
1233:k
1231:2
1225:k
1220:)
1218:θ
1214:r
1201:k
1176:)
1170:π
1166:a
1158:a
1149:2
1146:/
1142:π
1134:θ
1129:)
1123:a
1119:r
1109:)
1103:a
1095:a
1085:θ
1080:)
1074:a
1070:r
1046:k
1035:7
1032:/
1029:1
1019:5
1016:/
1013:1
1003:3
1000:/
997:1
990:k
981:4
978:/
975:1
967:k
959:6
956:/
953:1
944:π
940:T
936:π
934:8
926:2
923:/
920:1
912:k
904:4
901:/
898:1
889:π
885:T
881:π
879:4
871:2
868:/
865:1
857:k
849:π
845:T
839:π
837:2
832:π
830:2
825:π
823:4
819:T
814:.
811:a
807:r
799:r
792:π
788:r
783:.
781:k
777:a
765:4
762:/
758:T
750:θ
743:4
740:/
736:T
730:−
726:)
720:a
716:r
709:a
707:(
698:r
688:k
684:/
680:π
669:2
666:/
662:T
651:r
641:k
637:/
633:π
631:2
624:T
582:k
578:θ
574:r
570:a
562:k
538:k
536:2
533:/
529:π
520:)
514:a
510:r
505:)
499:a
495:r
488:.
475:)
470:)
463:k
460:2
444:(
440:k
436:(
426:=
422:)
416:2
402:k
398:(
388:=
385:)
379:k
376:(
340:)
334:(
325:)
319:k
316:(
307:a
304:=
301:)
295:(
286:r
283:=
276:y
269:)
263:(
254:)
248:k
245:(
236:a
233:=
230:)
224:(
215:r
212:=
205:x
175:)
169:k
166:(
157:a
154:=
151:r
82:k
78:)
72:r
60:d
56:/
52:n
44:k
39:)
33:r
23:.
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