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The Moon follows an arc across the sky that nearly completes a loop around Earth in about one day, while its illuminated face changes shape over a cycle of days. You can see both effects directly in Bartosz Ciechanowski's interactive "Moon" article (published December 17, 2024), which lets you drag the view, scrub a date-and-time slider, and watch the Moon's position and phase update in real time. The explanation below covers what those simulations show and what to look for.

The daily arc: one loop, almost

Over the course of a single day, the Moon travels on an arc across the sky and almost completes a loop around Earth. In the article's panorama simulation, you drag the view to change your viewing direction — this reveals the sky both above and below the horizon — and move sliders to see how the Moon's position shifts across days and hours of your local time. A small arrow points to the Moon's location as it moves.

Two practical details from the simulation:

• You can drag a figurine on a globe in the bottom-right corner to see how the sky looks from any location on Earth.
• If your browser allows it, a button places the figurine at your current location automatically.

The "almost" matters: because the Moon is also progressing in its orbit while Earth rotates, it returns to roughly the same spot a bit later each day rather than exactly one rotation later.

Why the appearance changes: phases over days

As days pass, the Moon's illumination visibly changes. The article notes that focusing on the tiny Moon as it shifts position is hard, so it zooms the camera in and locks it on the Moon. From that locked view, the phase cycle becomes the dominant visible change, separate from the daily motion across the sky.

Rotation and wobble: why we sometimes see the edges

In the zoomed, locked view, two subtler motions appear:

• Across a single day, the Moon seems to rotate.
• Over many days, it quite visibly wobbles.

These wobbly variations — the effect known as libration — let observers on Earth occasionally see some hidden parts on the "edges" of the Moon. Despite this, the Moon ultimately shows us only one of its sides.

What you can and can't see from Earth vs. space

The article frames this contrast directly: in the space demo you have freedom of motion, but "we don't have that freedom of motion in our daily experience" — the Moon wanders on its own path across the daily and nightly skies.

How to use the simulations

1. Space view: Drag the Moon to change your point of view; use the slider to control date and time. Expect to see the full sphere, including the far side.
2. Sky panorama: Drag the panorama to adjust viewing direction, including above and below the horizon. Move the sliders to advance days and hours in your local time. Expect the Moon's position to shift, with the arrow marking it.
3. Location: Drag the figurine on the globe (or use the auto-locate button if your browser permits) to see the sky from a chosen place on Earth.
4. Zoomed, locked view: Let the camera zoom in and lock on the Moon. Expect to see the daily apparent rotation and the multi-day wobble, plus the changing illumination.

What to take away

The Moon's behavior splits into three observable layers: a daily arc that nearly loops Earth, a phase cycle that changes its illuminated shape over days, and a slow wobble that occasionally reveals edge regions. Ciechanowski's interactive format makes each layer separable — you can lock the camera to isolate one effect at a time, which is difficult when watching the actual sky.

The Columbia University Mathematics Department is the mathematics faculty and academic unit housed in the Mathematics hall on Columbia's Morningside campus in New York. It offers an undergraduate program, a graduate PhD program, and a Master of Arts in Mathematical Finance, and it supports research across areas including algebraic geometry, geometry and analysis, mathematical physics, number theory, probability and financial mathematics, and topology. If you are a prospective student, a current student looking for course or tutoring information, or a researcher checking seminar schedules, the department's site is organized around those needs.

Where the department is and how to reach it

The department is located in the Mathematics hall on the Morningside campus of Columbia University in New York. The site's Quick Links and contact pages point to the main entry points:

• Contact Us
• Faculty Directory
• Calendar
• Seminar List
• Placement and Transfer Credit
• Calculus Classes
• Course and Exam Schedules
• Tutoring Services

If you need to reach a specific person rather than the department generally, the People section includes a Department Directory, an alphabetical faculty listing, faculty by rank, an alphabetical student listing, students by year, a staff directory, and a "Who Does What" page, plus office hours.

Research areas

Research activity in the department is organized around several broad areas listed on the site:

• Algebraic Geometry
• Geometry and Analysis
• Mathematical Physics
• Number Theory
• Probability and Financial Mathematics
• Topology

These labels matter if you are choosing a department for graduate study or looking for a seminar to attend: they indicate where faculty and student seminars cluster, and the site's seminar listings are the practical way to see what is running in a given term.

Degree and program options

The department's Programs section covers three main paths:

The MA in Mathematical Finance is run as its own program with a dedicated website, so admissions and curriculum details for that degree sit outside the main department pages. The PhD program is supported by Graduate Core Courses and Graduate Topics Courses, with course and exam schedules published separately.

Courses, placement, and tutoring

For students working through the course sequence, the site separates a few distinct tasks:

• Calculus courses: the Calculus Classes page covers placement information and sample syllabi for various levels of study. This is the page to check before registering if you are unsure which calculus level fits your background.
• Course and exam schedules: published under Courses, alongside an overview of Graduate Core Courses and Graduate Topics Courses.
• Placement and transfer credit: handled through its own Quick Link, relevant if you are bringing credit from another institution.
• Tutoring Services: listed both under Quick Links and Departmental Resources.

The site also lists a Mathematics Library, a Computer FAQ, Cantor Lounge, and Help Rooms as departmental resources.

Seminars, events, and calendars

The department runs numerous faculty and student seminars on a weekly basis throughout the academic year. To find them, the site provides:

• Calendar
• Weekly Seminars
• Department Colloquium
• Upcoming Conferences and Special Lectures
• Archived Seminars, Lectures, and Conferences
• Seminars at Nearby Institutions
• Academic Calendar

If you are outside Columbia and want to attend, the seminar list and calendar are the starting points; "Seminars at Nearby Institutions" is useful if your interests sit between departments or campuses.

Alumni, giving, and community

For people who have already passed through the department, there is an Alumni section with Information for Alumni, Events for Alumni, PhD Placement Data, and Recent Doctoral Theses. PhD Placement Data is worth checking if you are evaluating the doctoral program's outcomes. There is also a Giving page, and a Community Engagement section that includes the Association for Women in Mathematics.

Practical starting points by goal

• Prospective undergraduate: Programs → Undergraduate Program, then Calculus Classes for placement and Course and Exam Schedules.
• Prospective PhD applicant: Programs → Graduate PhD Program, plus PhD Placement Data and Recent Doctoral Theses to judge outcomes and research topics.
• Mathematical Finance applicant: go directly to the Math Finance website linked from the Mathematics of Finance section.
• Current student needing help: Tutoring Services, Help Rooms, Course and Exam Schedules, and Placement and Transfer Credit.
• Visitor or researcher: Calendar, Weekly Seminars, Department Colloquium, and the Faculty Directory.
• Alumnus: Alumni Information, Events for Alumni, and Giving.

One caveat: the site lists available faculty positions and reimbursement guidelines, but it does not publish tuition, fees, or program costs on these pages, so budget questions need to be directed to the relevant program rather than inferred from the department site.

Tiger Algebra is a free online algebra solver and calculator that shows step-by-step solutions without requiring a download or signup. It runs in a mobile or desktop browser, so you can use it to check homework answers, study the steps behind a solution, or quickly work through common algebra problems. You enter an equation or problem, press Solve, and the site returns a solution with steps.

What Tiger Algebra covers

The homepage lists a broad set of solvers and topics. Based on the site's own listing, these include:

• Order of operations — e.g. 2+3*4, (8+4)/3
• Least common multiple (LCM) — e.g. lcm(12,20), lcm(18,24,30)
• Greatest common factor (GCF) — e.g. gcf(12,18), gcf(4,8,32)
• Linear equations with one unknown — e.g. 2x=8, 3x+2=7x-5
• Percentages — e.g. 20% of 80, 15 is what % of 60
• Factorials — e.g. 5!, 8!/5!
• Quadratic equations using the quadratic formula
• Linear equations with two unknowns
• Polynomial root calculator
• Scientific notation
• Simplifying radicals
• Factoring trinomials
• Polynomial long division
• Reducing fractions to lowest terms
• Arithmetic sequences
• Trigonometry — exact values, unit circle, and graphs

The site also shows a running count of problems solved (over 20 million at the time of the source page) and a "Latest Solved" feed that includes inequalities, distance and midpoint problems, and limit-style expressions such as slim(x→0)(sin(x)).

How to use it

1. Enter your problem. Type an equation or expression into the input field on the calculator page — for example 2x=8 or lcm(12,20).
2. Press Solve. The site processes the input and returns the answer with step-by-step solutions.
3. Use Learn, Practice, or Quiz. Each topic row on the homepage links to these modes, so you can read an explanation, work through practice problems, or test yourself.
4. Try camera input if needed. A camera input option is present, but the page notes that camera input is not recognized, so typing is the reliable path.

Expected result: a solved answer plus the steps used to reach it, which you can follow to understand the method rather than just the final number.

What it's good for — and its limits

Good fit:

• Checking your own work on standard algebra problems
• Seeing intermediate steps when you're stuck
• Quick calculation of LCM, GCF, factorials, percentages, and similar operations
• Practicing with the Learn/Practice/Quiz structure

Keep in mind:

• The source page does not state pricing, so free use is described by the site's own "free" label; no payment details are given.
• Camera input is listed but flagged as not recognized, so don't rely on it.
• The solver list covers the topics shown above; for problems outside that set, results may be limited.

If you want a no-signup way to get step-by-step algebra help in a browser, Tiger Algebra fits that use. Start by typing your equation and pressing Solve, then use the Learn and Practice links to go deeper on any topic you need.