I don’t consider myself an artist but I always appreciate art. For me, art is way beyond painting or music: it is embedded in how we do things we love. I know, I know, it may sound cheesy but this TEDx video from 3Blue1Brown (and almost every single one of his videos) somehow convinced me that art could exist in unexpected entities, from photograph to math, or even — just like the mosaic above — a weird combination of both.
Hi, I’m Albers…
Let ABC be a triangle. Let BE and CF be internal angle bisectors of ∠B and ∠C respectively with E on AC and F on AB. Suppose X is a point on the segment CF such that AX is perpendicular to CF, and Y is a point on the segment BE such that AY is perpendicular to BE. Prove that
To answer these questions, we need to go back to the era of Euclid when he created the Elements. The first book of the Elements contains 5 postulates. We won’t bother the first four and only focus ourselves on the fifth and the famous Parallel Postulate:
ABCD is a parallelogram with a line g passing point A. Prove that the distance from C to g is either the sum or the difference of the distance from B to g and the distance from D to g.
This exercise uses Similarity. Please kindly read Theorem 1.1 beforehand if you’re not familiar with it.
Let P, Q, R be the feet of perpendicular from B, C, D, respectively to g. We will consider 4 cases:
Before reading this article, you’re encouraged to read the previous one just to give a little bit of context to what we are gonna do. This article examines two problems: dimensionality reduction and clustering. Enjoy!
In the previous article, we were performing classification on the Human Activity Recognition dataset. We know that this dataset has so many features (561 to be exact) and some of them strongly correlate with each other. Random Forest model can classify human activities as good as 94% accuracy using this dataset. However, it takes forever to do so. The second candidate is k-NN model with…
In mathematics, dimensions are parameters used to describe the position and characteristics of an object in space. I think we all agree that a 0-dimensional object is a point, which has no length, height, or depth. Moreover, a 1-dimensional object is simply a line. But is that it? Are there any 1-dimensional objects other than lines?
Before going on further to solve more geometry problems, we should settle this down. Given two points A and B, in the Euclidean sense, we can trace the shortest distance between the two as shown below.
This article uses Similarity. Please kindly read Theorem 1.1 beforehand if you’re unfamiliar with it.
In this article and many more to come, we will discuss some unique properties of triangles. Given an arbitrary triangle, define the followings:
Let ABCD be a parallelogram. On the outer side of the parallelogram, four squares are constructed: ABC₁D₁, BCD₂A₂, CDA₃B₃, and DAB₄C₄. On the outer side of triangles AB₄D₁, BC₁A₂, CD₂B₃, and DA₃C₄, four more squares are constructed having B₄D₁, C₁A₂, D₂B₃, and A₃C₄ as one of their sides and centers
respectively. Prove that
Hello! Welcome to my first article ever on Euclidean geometry. This article and many more to come are beginner-friendly and hence appropriate for those who are even on their first-day learning. We will start with the very basic concept of high school Euclidean geometry and build our understanding throughout time. At the end of the day, hopefully, we can tackle some of the hardest olympiad questions using only our established theorems.
To preserve continuity in reading and simplify future references, we will caption every figure and theorem by numbers. For example, in this article, we have Figure 1.1 and Theorem…
A modern smartphone is equipped with sensors such as an accelerometer and gyroscope to give advanced capabilities and facilitate a better user experience. The accelerometer in a smartphone is used to detect the orientation of the phone. The gyroscope adds an additional dimension to the information supplied by the accelerometer by tracking rotation or twist.
There have been studies conducted to utilize these sensors, such as in estimating road surface roughness conditions. However, what we’re doing will be more similar to this study by Harvard University. …