http://www.goodreads.com/quotes/54481-far-out-in-the-unchart...
https://en.wikipedia.org/wiki/File:Artist%27s_impression_of_...
The Earth is on that little strip at the bottom center of that picture, which looks like it's crossing between two of the larger arms.
Because the galaxy is disc-like and Earth is located about halfway between the center of the disc and it's edge, we're not in a great place to get a good perspective of how the galaxy actually looks. We basically watch stuff spin around us and try to estimate what it looks like from a top-down view.
The study concludes that parts of what we previously attributed to other arms actually belong to the arm that Earth is located on, making the arm longer than previously estimated.
http://astronomy.stackexchange.com/questions/10175/how-many-...
Here's one of my favourite composites. Check out the 4000 x 1212 pixel version.
https://en.wikipedia.org/wiki/File:Milky_Way_Arch.jpg
And here's one of my favourite non-composites:
http://apod.nasa.gov/apod/ap150730.html
Finally, here's a small patch of sky with a lot of various disk-like galaxies at random face-on-vs-edge-on tilts.
https://en.wikipedia.org/wiki/File:Hubble_ultra_deep_field_h...
Long exposure times, carefully stabilized equipment, and even moving the equipment outside the atmosphere have been a boon in producing these types of images.
We can't look at the Milky Way but we can look at similar galaxies. Our current understanding of what our own galaxy looks like is based on measuring the shape and direction of (groups of) stars. Stars contain(ed) Hydrogen and Hydrogen emits a radiation with a specific wavelength, that we can use to detect it. With Doppler Shift we can measure direction, if the emitting atoms are moving away, the wavelength is slightly larger. And they'll be slightly less if they're moving towards us.
Stars orbit the center of our galaxy and if we can get their orbit radius we can calculate how fast they are going. With enough measurements in all directions and using a lot of trigonometry, we can get to a map.
https://en.wikipedia.org/wiki/Cosmic_distance_ladder
which includes the methods you described.
Parallax. Telescopes.
https://medium.com/starts-with-a-bang/how-do-we-know-the-dis...
"the same math that enables you to hold your thumb at arm’s length, close one eye and then switch eyes and watch your thumb appear to shift, allowed us to measure the distances to the stars."
"Known as parallax, the fact that our planet’s orbit is some 300 million kilometers in diameter around the Sun means that if we view the stars today versus six months from now, we’ll see the closest stars appear to shift position in the sky relative to the other, more distant stars."
There's more in the link above. Then...
Satellites.
http://news.nationalgeographic.com/2016/09/gaia-milky-way-ma...
"For thousands of years, astronomers from the ancient Babylonians to Tycho Brahe had preoccupied themselves with noting the stars’ precise locations, a crucial foundation to understanding the cosmos. But the field sputtered in the 1960s, when scientists reached the smallest parallaxes that Earth-based telescopes could measure, stymied by interference from our rippling atmosphere.
It wasn’t until the 1980s and 1990s that the ESA satellite Hipparcos took astrometry to space, where it ultimately measured the precise distances of more than 100,000 stars. Gaia is even better: Hipparcos’s gaze reached only as far as 1,600 light-years away, barely leaving our celestial backyard, but Gaia is able to spy on stars up to 30,000 light-years away."
Remote controlled radio telescopes across the Earth, as in the article we comment to:
https://science.nrao.edu/facilities/vlba
"The Very Long Baseline Array (VLBA) is an interferometer consisting of 10 identical antennas on transcontinental baselines up to 8000 km (Mauna Kea, Hawaii to St. Croix, Virgin Islands). The VLBA is controlled remotely from the Science Operations Center in Socorro, New Mexico. Each VLBA station consists of a 25 m antenna and an adjacent control building. The received signals are amplified, digitized, and recorded on fast, high capacity recorders. The recorded data are sent from the individual VLBA stations to the correlator in Socorro."
"Precision astrometry is a VLBA science centerpiece. The relative astrometric accuracy of ~ 10 micro-arcseconds achievable with the VLBA is better than what the Gaia satellite is designed to achieve for most stars in its catalog (scheduled for release after 2015"
https://science.nrao.edu/facilities/vlba
"In 2010, the VLBA will begin a long-term program to determine the complete 3D structure of the Milky Way by measuring parallaxes with 10 mas accuracy or better to ~ 400 high-mass star-formation regions."
In order to achieve the stated accuracy using the radio telescopes around the world together is necessary.
This person seems to be using a non-standard definition of "math".