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1780 Reflections of a Polygonal Bird

/ ivasallay / Leave a comment

Today’s Puzzle:

What ordered pairs were used to create this bird?

Its eye was formed from an equation of a circle:
(x – 7)²+ (y – 15)² = 3/4.

After creating the polygonal bird using ordered pairs and that circle equation, I wanted to do other things with the bird. Everything I did was like a puzzle for me to figure out.

Could I make it “fly”? Yes!

 

Could I make it reflect itself more than once over the y-axis and the x-axis? Yes! And I could make it do some sliding at the same time!

This next one was the toughest for me to do. I wanted the bird to be in motion rotating counter-clockwise around the origin. I was able to do it, but Desmos wouldn’t save the sliders exactly the way I wanted. I will need your help on this one. Click on this rotating bird link, then push play on slider a. About the time that slider goes to zero, push play on slider b. If you hit the sliders just right, it will look something like this GIF I made, but slower:

Rotating Polygonal Birds

make science GIFs like this at MakeaGif

 

Factors of 1780:

Perhaps our polygonal bird would like to fly to a tree. Here’s a factor tree for 1780 that it can take a rest on.

I knew that 1780 was divisible by 4 because its last two digits are divisible by 4.

  • 1780 is a composite number.
  • Prime factorization: 1780 = 2 × 2 × 5 × 89, which can be written 1780 = 2² × 5 × 89.
  • 1780 has at least one exponent greater than 1 in its prime factorization so √1780 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1780 = (√4)(√445) = 2√445.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1780 has exactly 12 factors.
  • The factors of 1780 are outlined with their factor pair partners in the graphic below.

More About the Number 1780:

1780 is the difference of two squares in two different ways:
446² – 444² = 1780, and
94² – 84² = 1780.

1780 is the sum of two squares in two different ways:
42² + 4² = 1780, and
36² + 22² = 1780.

1780 is the hypotenuse of four Pythagorean triples:
336-1748-1780, calculated from 2(42)(4), 42² – 4², 42² + 4²,
780-1600-1780, which is 20 times (39-80-89)
812-1584-1780, calculated from 36² – 22², 2(36)(22), 36² + 22², and
1068-1424-1780, which is (3-4-5) times 356.

1780 is KK in base 88 because
20(88) + 20(1) = 20(89) = 1780.

1743 Finding Ways to Transform My Heart

/ ivasallay / Leave a comment

Today’s Puzzle:

An equation of a unit circle centered at the origin is x² + y² = 1.

If we change just the “y” part of that equation, we can get a lovely heart just in time for Valentine’s Day. Try it yourself by typing the equations into Desmos.

There are other mathematical equations for a heart, but this is the one I’m exploring in this post.

I was puzzled over how I could transform that heart. Can I make it bigger, or dilate it? Can I slide it away from the origin or translate it? Can I rotate it? Can I reflect it across the x or y-axis?

These are questions I’d like you to explore as well.

Heart Dilation:

In this first graphic, I was able to make my heart bigger. What kind of math let me do that? Also, how did I color the inside of some of the hearts? Look at the equations next to the heart and try to figure it out. The concentric hearts are evenly spaced. Do you recognize a pattern in the numbers that made that happen?

Heart Slide (Translation):

If we changed the center of a circle to (a, -b) instead of the origin, we would slide the whole circle. Here’s how we change the equation of the circle to give it a new center:
(x-a)²+(y+b)² = 1.

Similarly, in the next graphic, I was able to slide my heart away from the origin. How did I do that? Look at the equations to see how.

Heart Rotation:

A circle looks the same no matter how it is rotated, but the same isn’t true for a heart. Look at the equations below. How was I able to rotate my heart around the origin?

Heart Reflection:

Since a heart is symmetric, its reflection across the x-axis doesn’t look that interesting to me. Instead, I created a double heart that I reflected across both the x-axis and the y-axis:

Just for Fun:

Next, I was curious about what would happen if I changed the exponents on the outside of the parenthesis, so I changed a 2 from my original equation to an 8 in a couple of different places as I moved the heart from left to right. How did changing the exponent affect my heart? I found that as long as the exponent stays even, it still looks a little like a heart.

I was also curious about what would happen to my heart if I changed the “2/3” to a different fraction. I used fractions less than one as well as fractions greater than one. For many of my fractions, I used the post number, 1743, as the denominator. As long as the numerator was even and the denominator was odd, the graph still looked mostly like a heart. However, the closer the fraction was to zero, the more it looked like a circle.

Finally, I created this lovely flower using some of what I learned by making these transformations:

And for just a little bit more fun, I created a simple but chaotic-looking animation that I’ve titled Hearts in Motion. Enjoy!

I had so much fun exploring this heart in Desmos. Thank you for allowing me to share my excitement with you. Here’s a different Desmos heart created by a reader and shared with me. Click on it to make the heart beat:

 

Factors of 1743:

  • 1743 is a composite number.
  • Prime factorization: 1743 = 3 × 7 × 83.
  • 1743 has no exponents greater than 1 in its prime factorization, so √1743 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1743 has exactly 8 factors.
  • The factors of 1743 are outlined with their factor pair partners in the graphic below.

More About the Number 1743:

1743 is the difference of two squares in four different ways:
872² – 871² = 1743,
292² – 289² = 1743,
128² – 121² = 1743, and
52² – 31² = 1743.