... and I forgot to point out that there are actually 2 of these gadgets.
pic.twitter.com/L844x6BppE
This is the legacy version of twitter.com. We will be shutting it down on June 1, 2020. Please switch to a supported browser, or disable the extension which masks your browser. You can see a list of supported browsers in our Help Center.
You can add location information to your Tweets, such as your city or precise location, from the web and via third-party applications. You always have the option to delete your Tweet location history. Learn more
Add this Tweet to your website by copying the code below. Learn more
Add this video to your website by copying the code below. Learn more
By embedding Twitter content in your website or app, you are agreeing to the Twitter Developer Agreement and Developer Policy.
| Country | Code | For customers of |
|---|---|---|
| United States | 40404 | (any) |
| Canada | 21212 | (any) |
| United Kingdom | 86444 | Vodafone, Orange, 3, O2 |
| Brazil | 40404 | Nextel, TIM |
| Haiti | 40404 | Digicel, Voila |
| Ireland | 51210 | Vodafone, O2 |
| India | 53000 | Bharti Airtel, Videocon, Reliance |
| Indonesia | 89887 | AXIS, 3, Telkomsel, Indosat, XL Axiata |
| Italy | 4880804 | Wind |
| 3424486444 | Vodafone | |
| » See SMS short codes for other countries | ||
This timeline is where you’ll spend most of your time, getting instant updates about what matters to you.
Hover over the profile pic and click the Following button to unfollow any account.
When you see a Tweet you love, tap the heart — it lets the person who wrote it know you shared the love.
The fastest way to share someone else’s Tweet with your followers is with a Retweet. Tap the icon to send it instantly.
Add your thoughts about any Tweet with a Reply. Find a topic you’re passionate about, and jump right in.
Get instant insight into what people are talking about now.
Follow more accounts to get instant updates about topics you care about.
See the latest conversations about any topic instantly.
Catch up instantly on the best stories happening as they unfold.
... and I forgot to point out that there are actually 2 of these gadgets.
pic.twitter.com/L844x6BppE
Right! So now the count is correct. Le me see what Sage makes of it…
Assuming the two fixed points have coordinates (0,0) and (0,−2), the teardrop's equation is: x⁴ + 2·x²·y² + y⁴ + 10·x² − 6·y² + 8·y − 3 = 0. The sage commands used to perform the computation are here:https://gist.github.com/Gro-Tsen/35c2e3ed4ac8a3b04229f489f5b195df …
And here is the teardrop's shape computed from this equation:pic.twitter.com/8OU5XadIjT
That's a much neater equation than I expected! I wonder if there's a nice parametric form using elliptic functions à la https://arxiv.org/abs/1501.07157 (see e.g. Lemma 3.11)
Well, one can compute the curve's genus. I'm a bit too tired to attempt this right now. Another thing would be to understand why the equation given by elimination theory had another component (a circle with center (0,−1) and radius 2): this is probably obvious, but IDC.
The other component arises if you "flip" the smallest rhombus, collapsing the blue point to one of the other joints.pic.twitter.com/Jaew3aF7Vo
And unless I made a typo, the curve seems to be genus 3, unfortunately: x,y,z = PolynomialRing(QQ, ['x','y','z']).gens() qc = QuarticCurve(x^4 + 2*x^2*y^2 + y^4 + 10*x^2*z^2 - 6*y^2*z^2 + 8*y*z^3 - 3*z^4) qc.genus() ## result: 3
But now I'm a bit confused (and out of my depth): qc.geometric_genus() ## result: 0
Yes, that's the sort of thing I feared: there's probably a lot of fine print in Sage's genus computing commands about what it computes exactly and how. (Me, I can't even ever remember which is which between geometric and arithmetic genus, so…)
I believe the geometric genus is the one that matters re: parametrization; for some reason the QuarticCurve.genus() method gives the arithmetic genus (contradicting the behavior of Curve.genus(), of course...). Anyways, it turns out Sage has Curve.rational_parametrization() ...
And the output is: Scheme morphism: From: Projective Space of dimension 1 over Rational Field To: Projective Plane Curve over Rational Field defined by x^4 + 2*x^2*y^2 + y^4 + 10*x^2*z^2 - 6*y^2*z^2 + 8*y*z^3 - 3*z^4 ...
Defn: Defined on coordinates by sending (s : t) to (54*s^4 - 36*s^2*t^2 - 16*s*t^3 - 2*t^4 : -63*s^4 - 84*s^3*t - 26*s^2*t^2 - 20*s*t^3 + t^4 : 45*s^4 - 12*s^3*t + 22*s^2*t^2 + 4*s*t^3 + 5*t^4)
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.