iOS 8 supports widgets in the Notification Center. Apps that are properly built will be visible there and support a subset of the app's full features, often for quick reference. Here are four science oriented apps currently shipping that integrate with Notification Center.
1. Sky Guide. This is an astronomy and sky chart program that won an Apple 2014 Design Award. The app itself is similar to Star Walk and SkySafari, and in the Notification Center it displays a useful subset of information: sunrise, sunset, moonrise and moonset by default.
The graphic shows the time of daylight and darkness for your location. You can also bring up rise and set information for the major planets visible with the human eye. The curved lines are color coded to the sun, moon and the planets and show the relative altitude while above the horizon. You can show the sun and moon alone or "Show More..." to see rise and set time for all the major planets."
2. The Weather Channel. You may already have the latest version of The Weather Channel app on your iPhone. In the Notification Center, it shows your location, the time, temperature and near term text forecast.
Also included is a video that may or may not apply to your current location. I'm not a fan of that, and it represents, in my experience, a certain amount of overdoing things by this developer It requires great skill to give people good information with a great UI without distractions, this is a so-so effort, and so I'm not a big fan of this app anymore, nor its implementation in N.C.
The Weather Channel. Video is unwelcome, unnecessary.
3. Yahoo Weather. This is a better implementation in Notification Center than The Weather Channel's. There's a theme graphic that represents the current sky conditions plus temperature: current, high and low, plus a graphic of the forecast. That's not as complete as one might want, in terms of a short text blurb, but it's compact and may be all you need, especially in concert with Apple's own text forecast in the "Today Summary."
This is a nifty, minimalist presentation and has earned a permanent place in my Notification Center.
4. PCalc. In my opinion, this is the very best scientific and everyday calculator for iOS, and everyone should have a copy. It costs a little more than most, more than you might ordinarily pay for an app, but the value is considerable. The author, James Thomson, understands the numerical analysis of calculators, avoids amateur mistakes, and he's aggressive with iOS technology.
In the Notification Center, the developer brings up a simpler version of PCalc than you may have in your PCalc app settings, but he preserves the setting of RPN or Algebraic. What's nice about Notification Center is that you may have PCalc in a folder, and that may require a few swipes and a tap to launch. But you'll always have instant access to a first-class calculator in Notification Center. Mine is staying put in N.C.
PCalc (Not pi, rather an approximation: 355/113)
I expect many more apps to be released soon that have a science or technical basis and are integrated with Notification Center. I'll keep and eye out for them.