#Properties of Logarithm

[existinglimits]

[forgottenbones]

The iPhone of Slide Rules - Numberphile

[divinegastronomer]

The Golden Ratio

Incorporating the concept of the golden ratio (approximately 1.618) into diet planning is an innovative approach, blending aesthetics and numerical proportions with nutrition.

The golden ratio, a mathematical ratio commonly found in nature, art, and architecture for its aesthetically pleasing properties, can also be creatively applied to diet planning.

Here's a conceptual framework for how one might do so:

*Visual Proportions on the Plate

Apply the golden ratio to the visual composition of your meals. For example, you can aim to fill approximately 61.8% of your plate with vegetables, fruits, and whole grains, while the remaining 38.2% could be allocated to proteins and healthy fats. This not only ensures a balanced diet but also makes your meal visually appealing.

*Macro-Nutrient Ratios

You could also apply the golden ratio to your macro-nutrient distribution—carbohydrates, proteins, and fats. Although traditional dietary recommendations might not perfectly align with the exact golden ratio numbers, you can aim to get close. For instance, you might try to consume a larger proportion of your calories from carbohydrates and fats (combined in a way that mirrors the golden ratio) with a smaller proportion from proteins.

*Caloric Distribution Throughout the Day

Consider using the golden ratio to plan the distribution of calories across your meals throughout the day. For example, if your daily caloric intake is 2000 calories, you could plan to consume about 1236 calories (61.8% of your daily intake) by the end of lunch and the remaining 764 calories (38.2%) for the rest of the day.

*Ingredient Ratios in Recipes

Incorporate the golden ratio into recipe ingredient amounts. For instance, when making a smoothie, salad, or dish, consider the volume or weight of ingredients in relation to each other following the golden ratio. This could mean using 1.618 times as much of one ingredient as another, creating both a balanced and aesthetically pleasing dish.

*Planning Eating Intervals

If you’re into intermittent fasting or planning your meal times, use the golden ratio to plan the intervals between eating. For example, if you prefer a shorter eating window, you might fast for 16.8 hours and eat within a 7.2-hour window, mimicking the ratio in a more abstract manner.

Incorporating the golden ratio into your diet is more about the approach to balance and aesthetics rather than strict nutritional guidelines. It encourages a mindful and visually pleasing way of preparing and consuming food, which can enhance the eating experience and potentially lead to more balanced meal choices.

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Nature showcases the golden ratio in various forms, exhibiting its intrinsic patterns and designs that adhere to this mathematical principle. Here are some notable examples where the golden ratio is observed in nature:

* Spiral Patterns in Plants and Flowers: The arrangement of leaves, seeds, and petals in many plants follows the Fibonacci sequence, which is closely associated with the golden ratio. For instance, the spiraling pattern of seeds in a sunflower or the way pine cones are arranged can exhibit the golden ratio.

* Animal Bodies: The proportions of animal bodies often reflect the golden ratio. For example, the bodies of many species of fish and the spiral shells of mollusks (like the nautilus) follow this pattern. Even the positioning of facial features in some animals adheres to these proportions.

* Hurricanes and Galaxies: Spiral galaxies and the pattern of hurricanes showcase the golden ratio in their formation. The arms of spiral galaxies and the eye of hurricanes can exhibit spirals that mirror the golden spiral, which is a logarithmic spiral whose growth factor is φ, the golden ratio.

* Human Bodies: The human body has several examples of the golden ratio, from the proportions of the face and head to the ratios of limb segments and the structure of the fingers. For instance, the ratio of the forearm to the hand often approaches the golden ratio.

* Fruit and Vegetables: The way some fruits and vegetables grow or are structured can showcase the golden ratio. This can be seen in the way seeds are arranged in a fruit or the spiral patterns of growth in vegetables like romanesco broccoli.

The golden ratio is not a strict universal law governing natural forms but rather a recurring pattern that shows how efficiently nature organizes structures and systems. Its prevalence suggests an aesthetic and functional efficiency in biological forms and processes, a testament to the interconnectedness of mathematics and the natural world.

*****

Incorporating the golden ratio into cooking and food preparation through an esoteric culinary philosophy involves integrating the harmony and balance found in nature into your diet. This approach can add a layer of mindfulness and intention to the act of cooking and eating, aiming to align with the natural patterns for health and well-being.

Understanding the Golden Ratio

The Golden Ratio (Φ = 1.618…) is about balance and proportion. In culinary terms, this can translate to balancing nutrients, flavors, colors, and even the way food is plated.

Ingredient Proportions

- Balanced Meal Composition: Use the golden ratio to guide the proportions of different components on your plate. For example, making the ratio of vegetables (1.618) to proteins (1.0) in your meals, aiming to balance the nutrient density in favor of whole, plant-based foods.

- Recipe Formulation: Apply the ratio to the quantity of ingredients. For instance, in a salad dressing, use 1.618 parts oil to 1 part vinegar, or apply it to the ratio of spices to achieve a balance of flavors.

Food Pairing and Plating

- Visual Harmony: When plating your food, use the golden spiral as a guide for arranging the components attractively. Placing the main item and accompaniments in a way that visually represents the golden spiral can make the meal more appealing.

- Harmonious Pairings: Choose ingredients that complement each other not just in flavor but in nutritional content, aiming to create dishes where the sum is greater than its parts, adhering to a holistic balance.

Mindful Cooking and Eating

- Intentional Preparation: As you cook, focus on the process as a form of meditation, contemplating the natural beauty and balance in the ingredients and dishes you are preparing.

- Conscious Eating: Eat slowly, savoring each bite, and consider the nutritional balance and harmony in your meal. This mindful approach can enhance digestion and satisfaction.

Personal and Seasonal Harmony

- Align with Nature: Incorporate seasonal and locally sourced ingredients, aligning your diet with the cycles of nature. This not only supports sustainability but also connects you to the natural world and its rhythms.

- Adaptation for Health: Use the principles of the golden ratio flexible to suit your health needs and goals, ensuring that the concept of balance is also about what is harmoniously beneficial for your body.

Culinary Creativity

- Innovative Dishes: Let the golden ratio inspire you to create new recipes or tweak traditional ones, finding a balance between innovation and tradition, between simplicity and complexity.

Using the golden ratio as a guide in your culinary philosophy is more about adopting a mindset of balance, harmony, and connectivity with nature than about following strict numerical rules. It encourages a thoughtful, creative, and holistic approach to diet, aiming to nourish both the body and spirit.

*********

Adopting an esoteric culinary philosophy that incorporates the golden ratio into cooking and food preparation could offer several health benefits, primarily through fostering a deeper connection with food and promoting a balanced, mindful approach to eating. Here are some potential benefits of this approach:

1. Improved Nutritional Balance: By focusing on the proportions and combinations of foods that mirror the harmony found in nature, you might naturally lean towards a more varied and balanced diet. This can enhance your intake of essential nutrients, supporting overall health.

2. Mindful Eating Practices: This approach encourages mindfulness in the selection, preparation, and consumption of food. Mindful eating has been linked to better digestion, a healthier relationship with food, and possibly aiding in weight management.

3. Enhanced Appreciation for Food Quality: An emphasis on the natural aesthetics and ratios in food preparation can lead to a greater appreciation for high-quality, unprocessed ingredients. This could reduce the consumption of processed foods, potentially lowering the risk of chronic diseases.

4. Encouragement of Creativity and Experimentation: Engaging with food on a deeper, more esoteric level can stimulate creativity in cooking, which could make healthy eating more enjoyable and sustainable over the long term.

5. Stress Reduction: The act of preparing food with intention and a focus on balance may have meditative qualities, potentially reducing stress levels. Lower stress is linked to various positive health outcomes, including lower blood pressure and improved mental health.

6. Social and Cultural Connection: This approach can also foster a deeper connection to the food traditions of various cultures that have historically embraced the golden ratio and other natural harmonics in their culinary practices, possibly enriching your eating experience and cultural understanding.

7. Detoxification and Digestio: A diet that emphasizes fresh, natural foods and harmonious combinations may support the body's detoxification processes and improve digestion, contributing to overall vitality and well-being.

[ophizz]

Ramblings of a Lunatic - 3rd Quarter reflection🌝 🌸

⚠️W A R N I N G⚠️

Firstly sir, before you subject yourself to reading this reflection of my learning journey, I would like you to know that you may need to take some of the information with a grain of salt. I am an over dramatic person sir, and I may have over dramatised my experiences. I apologise in advance for whatever I am about to write. I was not built for pisay, nor did I actually ever want to go through this harsh academic plan (or however you call it. training??). Thank you for being our teacher sir, thank you for your patience, I hope mag skip ka through a lot of parts, FYI boring siya sir

a. How would you describe your Math 3 second quarter learning journey?

It could be easily described with 3 simple words. I Give Up. I have given up sir, I'm not smart, I'm not hardworking, I'm not good, so it was only natural for my course in life to give up. No matter how many times the topic was discussed, no matter how many times I tried and redo all the problems, my brain can't handle it. My brain is unable to physically store it within its cells. “Memory Full, only 0.2 megabytes left” type of situation if you get me sir, like when my phone can't open WuWa because it takes up too much space. That's me in math, my brain can't run the math application  because the memory is full, and math takes up too much space. Adding onto that, I'm not prepared for the LT, nor have I passed the graphing activity you gave us sir. Further proving my point of giving up entirely on Math 3. It's not you sir i promise, it's a me problem that I'm too lazy to fix. 

b. Which topic did you find most enjoyable? What made it enjoyable for you? Provide clear

The topic I found most enjoyable was the easy ones. I felt like I was going on the right track but apparently it's like a roller coaster. At first it was fine and dandy, but as time went on, you could slowly feel the dread as it builds up inside. Then boom you're going up, down, left, right, and side to side, while your brain tries to grasp onto something to stabilise itself but whoopsies, apparently the handlebars broke. When you get off the ride, you tell yourself at least you enjoyed the beginning. The easy, calming, joyful part of the ride. To me, that part of the Ride was us learning the basics, the exponential to logarithmic and vice versa, as well as the properties of logarithm. To me, those were the best times of the helling ride. 

c. What concepts did you find easy to learn? What do you think made them easy for you?

The topics are the same as the previous question. I think they were easy for me because it only involved common sense, minimal memorisation, and simple arithmetic. Honestly sir, that's all my brain could handle. My brain overheats when it has too many things to do, so when we went to the solving parts of the later topics... thats when my processor got weaker. So basically I found the topics, exponential to logarithmic (vice versa) and properties of logarithm easy topics.

🌸♥*♡∞:｡.｡ P h o t o s ｡.｡:∞♡*♥🌸

Sir! I didn't say that my answers are correct sir🥺..... sorry sa photo dump po hehehe

d. What concepts did you find most interesting/inspiring? Why do you think so?

For me, the most interesting one was the compounded interest. I now know how to manage finances because of that topic as well as sir Mike's crash course on investing! In all honestly, it's because its the one with the closest correlation to real life use, unlike logarithms, or graphing. That's why I see it as the most interesting/inspiring. Especially when you want to invest in a Condo, or house for example, and you now know how to actually compute for the price and know how to compare to know where you save the most money in the long run. It prepares us for the future! :D

e. What concepts have you mastered most? Why do you think so?

I will mention again and again, exponential to logarithmic and vice versa. It is because it is the easiest, just simple arithmetic and you're done. I admit that my arithmetic may not be the best, but I think I can do the arithmetic for that specific topic sir.

˖ ᡣ𐭩 ⊹ ࣪ P H O T O S ౨ৎ ° ₊

f. What concepts have you mastered the least? Why do you think so?

Sir I have not mastered graphing. I am so sorry sir, but I did not understand your discussion sir... But I have an excuse! I was undergoing through the trials of satan. Pushed to my limits as a girl forced to face the consequences of not having a parasite growing inside my uterus. The burning pain of cramps and a migraine. Sir I'm so sorry I truly don't understand anything and I know there is no use for an excuse. That the excuse does not veil my stupidity for not listening and understanding the topic. Im sorry sir.

˖ ᡣ𐭩 ⊹ ࣪ P H O T O S ౨ৎ ° ₊

sir I actually have no photos to show you... I haven't even done the activity in google classroom. Im sorry sir... genuinely sir...

g. What quick notes do you have for:

i. your teacher;

Sir, Im sorry for not reading your messages properly and thoroughly...Sir especially when you asked about the competition sir... Sir I'm scared to apologise to you in person sir, but Im sorry for wasting your time po... Im sorry for all the things I have done that might have offended you sir, or annoyed you sir... Ill try my best to be a better person sir... Sir if I did anything mean or anything of that same nature sir, I promise it wasn't intentional sir... Im sorry sir....

ii. your classmates; and

I just now noticed that the water was boiling. Thankfully I got out immediately.

iii. yourself?

Maybe, I should give up. Sometimes it's okay to start all over again. Push your limits, but not too far. You could always work on yourself but you're just lazy. Thats all you are. Lazy. You will never amount to anything, humble yourself. No matter how hard you try, your work will never be appreciated. You will never shine in your family of stars. Know your place in life.

⚠️sir this is a joke⚠️

Maybe I should actually review for my subjects... maybe I don't try because Im scared that if I try nothing will improve. Im scared that if I try I would still amount to nothing.

I wish he was real and I could steal his black card. I could manage his finances with compound interest, trust! Sylus save me from Lucifer's infected urethra!

Sir, thank you for managing to read it all the way here if ever you did sir. Im sorry you had to read all of that. Sir, I hope you don't mind the fact that I am slowly going crazy over the length of this post. I hope you have a good day after looking at this submission sir... truly my sincerest apologies.

⋅˚₊‧ ୨Thank you for reading!୧ ���₊˚ ⋅

ଘ(੭ˊᵕˋ)੭ ੈ♡‧₊˚

[erbiumspectrum]

Hi! Do you have any tips for studying chemistry? For some reason I cant seem to get all the formulas in my brain.

Hey!

My unhelpful but still favorite advice for shoving formulas into one's brain is to understand them 😅 A purely memorization-based approach is very bad for chemistry.

If the problem seems to be particularly understanding/ remembering formulas:

Ask yourself if this particular formula is just words turned into numbers and mathematical symbols. I think it may not work for everyone, but for example I found it easier to remember the literal definition of pH that is "the negative decimal logarithm of hydrogen ion concentration" rather than "pH = -log [H+]" bc otherwise I'd keep forgetting about the minus sign.

Check if you find deriving a formula from another formula easier than just memorizing it. Again, my personal example is I hate memorizing things so much I never really bothered to remember the equation that describes Ostwald's law of dilution - bc I knew I could easily, quickly, and painlessly derive it from the equilibrium constant for concentration + degree of dissociation (and I've done it so many times now it stuck in my brain anyway).

When all else fails, I turn to mnemotechnics. To this day I remember that Clapeyron's equation goes pV = nRT because many years ago someone on the internet shared a funny sentence whose words start with these 5 letters. The sillier the better.

If the issue is with chemistry in general:

Take it chapter by chapter. Chemistry, like most STEM subjects, is just blocks of knowledge upon blocks of knowledge. For example, if you want to learn electrolysis, you need to understand redox reactions first. Try to identify where the struggle begins and work from there.

Once you've picked a topic you want to work on, follow the reasoning in your textbook. If you get stuck, that might be a sign you're simply missing a piece of information from a previous chapter. If an example comes up, try to solve it along with the tips in the textbook.

If anything remains unclear, it's usually not the best idea to just leave it and move on. If the textbook becomes unhelpful, turn to the internet or maybe a friend. Otherwise, the next chapter may just turn out to be needlessly confusing.

Practice problems practice problems practice problems!! And not just the numerical ones. The theory-based ones where they ask you about reactions, orbitals, the properties of the elements etc. are important too.

Choose understanding over memorizing whenever possible.

Try to look at the big picture: the way certain concepts are intertwined, how one law may be a logical consequence of another law you learnt before, why some concepts are taught together, why you had to learn something else first to get to what you're studying now. Again, as an example, I think it's particularly fun to see towards the end of ochem, somewhere around the biomolecules: you need to integrate your knowledge of aromatic compounds, ketones and aldehydes, alcohols, carboxylic acids... Stack new information upon what you already know.

Study methods I'm a big fan of: spaced repetition, solving past papers (anything I can get my hands on tbh), flashcards for the things I absolutely have to memorize, exchanging questions and answers with a friend, watching related videos.

If by any chance you end up taking pchem, I have a post for that specifically.

I hope you can find something helpful here :) Good luck!

[silenceitis]

An intuitive introduction to information theory, part 1

Motivation

Ok, let’s start with the basics; what exactly is information theory and why do we care about it?

As the name suggests, well, it’s the theory of information. More precisely, how do we “measure” how much “information” something has? What is the “best way” to represent this “information”? How can we communicate this “information”? How can we communicate information if there’s a chance some of it may be “lost” in transit?

As you may have guessed, this underpins many important technologies we all use today, such as digital communication and file compression.

All in all, it’s about developing a framework that allows us to quantify “uncertainty” and optimise communication.

Important ideas

Ok, so I’ve said the word “information” a lot. But what does this actually mean? Let’s think about the concept of surprise. If I told you “the sun will rise tomorrow”, you wouldn’t be very surprised would you? Or in other words, this statement wouldn’t give you much information. You are already pretty certain that the sun will indeed rise tomorrow. But what if I told you “your exam scheduled for 2 months time has been moved to tomorrow”? That would be pretty damn surprising right? Or, you could say it’s a very informative statement.

So, we can see that there is some relationship between the idea of surprise and the idea of information. Let’s try and describe this intuition a bit more precisely.

Clearly, how surprised we are about something is related to the probability of that thing happening. If I had a biased coin that had a 99% chance to land on heads, and 1% to land on tails, you’d be very surprised if I flipped a tails, and not very surprised at all if I flipped a heads. So surprise decreases as the probability of some event happening increases.

Further, if we have two completely independent things happening, it would make sense for our surprise to add. Going back to the same biased coin, suppose I flip it, and you feel whatever surprise you do about the result. If I flipped another biased coin at the same time, you would feel a certain amount of surprise about this too. But the amount of surprise you felt about the result of one of the flips wouldn’t change the amount of surprise you felt about the other, the two coins don’t affect each other do they?

And finally, if an event becomes slightly more likely, or slightly less likely, we would expect our amount of surprise to only change slightly too right? That is, the amount of surprise we feel about an event should depend continuously on the probability of the event.

If we denote s(A) to be our surprise for the event A, it turns out that S(A) = -log(p) satisfies all these properties, where p is the probability of A happening. Not only that, but this is actually the only function that satisfies these properties (up to our choice of base for the logarithm).

Ok, so we now have a sense of what we mean by surprise, and hence information, and an explicit function to calculate a numerical value representing this. But notice how to make use of this, we need to actually know what the outcome is. E.g. if I have a biased coin, you can’t tell me anything until I flip the coin and you interpret the result. But what if I have a coin and I haven’t flipped it yet? Based on the probabilities, we can work out how surprised we *expect* to be. Or in other words, on average, how surprised you will be.

This is exactly the concept of entropy!

Now, I’ve talked a lot about coin flips. You’re probably getting a bit bored of this. So let’s formalise things slightly. Suppose we now have a discrete random variable X, taking values in a set χ. As you may recall from any probability classes you’ve taken, the expected value of X can be calculated as

And we can calculate the expected value of a function of X as

Now, surprise is just a function of our random variable X, so we can use this formula to work out what entropy is! We will write entropy as H(X). Again, H(X) is just a value representing, on average, how surprised we are when we observe an event from X.

From this formula, we can say a few things about entropy.

Firstly, it it is always non-negative, which we expect; being “negatively surprised” about something doesn’t really make any sense right? So why would our average surprise ever be negative?

Furthermore, entropy is only zero when one of the probabilities in the sum is 1 (remember log(1) is zero, and probabilities sum to 1). Again, this makes sense. Our average surprise would only ever be zero if we were certain of what the outcome would be.

Ok great, we now know, given a random variable X, on average how surprised we will be. But what if we don’t actually know what the random variable X is? This happens all the time! In real life, we may be able to observe a random process, and hypothesise what we think the distribution is, but we may not know for certain. For instance, suppose I had a biased coin (yep, sorry, back to the coin again…), but I don’t tell you what the probabilities of landing on heads or tails are. You may take the coin, do a whole bunch of flips, and estimate the probabilities from your results. But your estimate is unlikely to be the true answer. We want to find a way to measure “how costly” it would be to use your estimate, rather than the true distribution. Or in other words, how much information we lose (or indeed surprise we gain) by using your estimate.

This leads us to the idea of divergence, a way of measuring how different two probability distributions are, by looking at the differences in entropy (aka surprise, aka information!) between the two.

Suppose we have a random variable with probability mass function p, on the set χ. But we want to encode X with a different pmf, q, also on χ. Then the average surprise when taking values from X and encoding with p (aka using p in our function S(A)) is just the entropy of X, H(X), as we discussed before.

And the average surprise when taking values from X but encoding with q (again, just using q in our surprise function) is

Notice how this is not quite the entropy of X, although it does look very similar. Our surprise has changed, as surprise is based on our “beliefs” of what the values of X will be, that is, it is based on our choice of encoding. But our “beliefs” do not change the actual probabilities of X taking particular values.

Hence, our divergence, written D(p || q), is just

However, the concept of divergence is not just used to compare how our guess for the distribution, q, compares with the true distribution, p. It can actually be used to compare how different *any* two distributions are, as long as they are both on the same set of values. More precisely, D(p || q) measures how different the two distributions are, *from the perspective of p*. I.e. we draw values from p, and measure how much more surprising that sample would be, on average, if it had come from q instead.

We do precisely the same thing as before, except instead of entropy, we have the slightly modified surprise function, as we did for q. Note that at no point have we made an assumption on what the actual distribution of X is here (which is why the sum is not the entropy of X, even if it looks very similar).

Now, as you may have noticed, what I said before hints at some asymmetry in the definition of divergence, and this is indeed the case. In general, D(p || q) is not equal to D(q || p)! If we think about the meanings of these two quantities, this makes sense. D(p || q) represents how “inefficient” it is to pretend that data from p came from q. Whereas D(q || p) represents how “inefficient” it is to pretend that data from q actually came from p. Intuitively, these are not the same thing.

Additionally, you may have noticed that the divergence can take the value infinity, when for some value of x, p is greater than zero, but q is zero. Intuitively, this means that q, the distribution you’re pretending the data came from, says that a certain event is impossible, but p says that this event *does* occur.

Ok, so now we’ve seen how we can “measure” how different one distribution is from another, which can be useful when we want to model an unknown distribution. But what if we don’t try to estimate our unknown distribution directly? Instead, we may be able to observe another random variable, and hypothesise that it may be able to tell us something about our unknown distribution.

This leads us to the idea of mutual information. How much does knowing Y tell us about X? Or in other words, how much is our uncertainty in X reduced by, if we also observe Y?

So really, we’re looking for a way of determining how much dependence there is between the two variables. If X and Y were independent, then knowing something about Y wouldn’t reduce our uncertainty in X. So we can measure how different the joint distribution of X and Y is, from the joint distribution if they were independent! This will give us exactly what we’re looking for; how “far” the joint distribution of X and Y is from being independent.

We can use our notion of divergence to calculate this! As you may recall, if X and Y are independent random variables, the joint pmf of X and Y is simply the pmf of X multiplied by the pmf of Y. So we just plug this into our divergence function, as below, writing I(X; Y) for the mutual information:

And with some rearranging and use of our previous definitions, we can get the following alternate forms for mutual information:

All of these definitions make it clear to us that mutual information is in fact symmetric, I.e. I(X;Y) = I(Y;X).

We can also see that mutual information is zero if and only if X and Y are independent, and that it is non-negative.

[crumblingspine]

I hope whoever invented the properties of logarithms knows that hell is a very hot place.

[miirshroom]

How Bloodborne astral clock investigation started:

We have the moon, a medical metaphor where the Great Ones are themed as body parts (thanks to Charred Thermos for delving into this), and a clock with 14 symbols split between 12 positions. Surely this has some structure following the 'Zodiac Man' that dictated bloodletting according to the moon in pre-Victorian times (and still shows up on websites that report info on the moon phases connected to calendar dates).

How it's going:

The Wheel of Logarius has 14 spokes which is reminiscent of the pH scale that goes from 1-14 and is a logarithmic scale based on concentration of H+ per litre of liquid. And given the medical context it would make a lot of sense that the zodiac is also split between the 4 Hippocratic humours of blood (spring), yellow bile (summer), black bile (autumn), and "phlegm" (winter). So "Formless Oedon" could naturally be "blood" which encompasses Aries/Iron, Taurus/Copper, and Gemini/Mercury. This actually makes sense in the context of Old One Coldblood being blue because that is the colour of horseshoe crab blood which has a copper metal core, but still falls under the domain of Oedon. And "quicksilver" bullets synthesized from blood have the fused property of real blood (iron based) and fantasy (hunting beasts with "quick" silver bullets). Saw something about a "flora, of the moon" which could be related to the traditional way of assigning a name to each moon instead of months. But which system? The Farmer's Almanac is a reliably documented, consistent, and accessible source but has the drawback that it is disconnected from the traditions of the native peoples. On the other hand there is the potential Osage connection with May being the 'Killer of the Flowers Moon'.

Also, the "third umbilical cord" would certainly be a body part, but I'm not so sure that's the only thing going on here. Bloodborne has a huge musical theme, and a "third chord" is a configuration of chord containing 3 tones at specific offsets from each other.

[existinglimits]

Learn your properties of logarithms

Just... just do it. Before you take calculus is best, but as soon as possible if you're in the middle of calculus.

You're not going to be able to do "logarithmic differentiation" if you aren't comfortable with log rules.

Below the cut are..

a link to Khan Academy Algebra 2 Unit 8: Logarithms

"Logarithms explained Bob Ross Style"

"What is the number 'e' and where does it come from?"

Link to Khan Academy Algebra 2 Unit 8: Logarithms

You can skip to "Properties of logarithms" if you're short on time, but most people who struggle with properties of logarithms do so because they don't understand logarithms.

Logarithms are the inverse of exponentials, so the rules are closely related but inverted. If you're REALLY struggling with logarithms, you may want to examine how comfortable you feel with properties of exponents.

Logarithms explained Bob Ross Style

What is the number "e"?

[hydrofarm]

Soil pH

What is the Meaning of pH? The term pH refers to the level of acidity or alkalinity of a substance. It’s based on a scale that ranges from 0 to 14, where a pH of 7 is considered neutral. Values below 7 signify acidity, while values above 7 indicate alkalinity. The pH scale is logarithmic, meaning each whole number change represents a tenfold difference in acidity or alkalinity. In simpler terms, a soil pH of 5 is ten times more acidic than a pH of 6.

Healthy soil pH is the foundation of a thriving garden. Find the balance, and your plants will flourish.

Understanding pH is important in soil science because it influences many soil properties, including nutrient availability, microbial activity, and plant growth. A proper balance of pH levels helps plants access nutrients more efficiently, making it an essential factor in successful gardening and farming.

What is the Best pH for Soil? The ideal pH range for most plants is between 6.0 and 7.0, which is slightly acidic to neutral. Within this range, most nutrients are readily available to plants. However, some plants have specific pH preferences. For instance, blueberries, azaleas, and rhododendrons thrive in more acidic soils with a pH between 4.5 and 5.5, while some ornamental plants and vegetables can tolerate slightly alkaline soils up to a pH of 7.5 or even 8.0.

How to Check pH in Soil? There are several ways to measure soil pH, ranging from simple DIY methods to professional testing:

pH Test Strips: You can purchase these at most garden centers. Simply mix a soil sample with distilled water, dip the test strip, and compare the color change to a pH chart.

Soil pH Meters: These electronic devices are quick and easy to use. Insert the probe into the soil, and the meter will display the pH reading.

Professional Soil Testing Kits: These kits usually come with detailed instructions and provide more accurate results than DIY methods. Some local agricultural extension offices and labs also offer soil testing services for a fee, which can give you a complete analysis of soil pH and nutrient levels.

What pH is Rich Soil? Rich soil typically has a balanced pH range between 6.0 and 7.5. At this level, soil can retain essential nutrients like nitrogen, phosphorus, and potassium, which are crucial for plant health and growth. Rich soil with an optimal pH level supports a diverse range of plants, from vegetables to flowering perennials, as it creates a favorable environment for beneficial bacteria and earthworms that break down organic matter and enrich the soil structure.

What is High pH in Soil? When soil has a pH above 7.5, it’s considered alkaline. High pH soils can pose challenges, as they may limit the availability of essential nutrients like iron, manganese, and phosphorus. Alkaline soils often appear in areas with low rainfall or where the bedrock contains limestone or other alkaline minerals. Plants in high pH soils can sometimes show signs of nutrient deficiencies, such as yellowing leaves (chlorosis) due to iron deficiency. While some plants, like lavender and certain succulents, can tolerate or even prefer alkaline soils, most garden plants prefer slightly acidic to neutral conditions.

What pH Level is Healthy for Soil? A healthy soil pH typically falls between 6.0 and 7.5, though this can vary depending on the type of plants you’re growing. Maintaining a balanced pH level ensures that your plants can efficiently absorb nutrients, promoting healthy growth and resistance to diseases. Soil pH can fluctuate over time due to factors like rainfall, fertilizer use, and plant root activity. Testing your soil pH regularly can help you stay on top of any changes and adjust your soil management practices accordingly.

You can use soil amendments to adjust the pH level of the soil. For example, lime can raise pH for overly acidic soils, while sulfur can lower pH in alkaline soils. However, changing soil pH is a gradual process, and it’s best to make adjustments in small increments and monitor the results.

Conclusion Soil pH is a critical factor for plant health, as it affects nutrient availability, soil structure, and microbial activity. Aiming for a pH range between 6.0 and 7.0 can support the growth of a wide variety of plants and contribute to a thriving garden or farm. Regular soil testing can help you monitor pH levels and make informed decisions to maintain a healthy balance. By understanding and managing your soil's pH, you can create an environment where your plants can truly flourish.

[joompheart]

Andrew in the Epilogue (I've got a "main story" for these guys, then a couple designs for the characters later after the final fight).

She gets just fucking mutilated in the final fight with Teddy, her entire body covered in gnarly burn and scar tissue. She takes so much damage that she is appreciably smaller when she finally heals back to functionality. The lost bone and muscle and organ tissue make her about half a head shorter and (if you can believe it) even skinnier. For anyone keeping track, she's gone from disgustingly skinny to revoltingly skinny. Her magical resilience allowed her to heal back to full functionality, but not full strength by a long shot.

More description under the cut.

Notably, the only part of her body not torn to hell is her right breast, just like Theodore. It's slightly clipped at the top but her tiny boob is otherwise untouched. The other breast was ruined along with the rest of her body.

Andrew is still strong for her size, but big chucks of flesh missing from her muscles means that she's meaningfully weaker than she was (sans magic). Her sword was also cleanly snapped by Teddy in their fight, removing the ability to modulate it's length. It's still longer than could physically fit in the sheath, but it's always that size when unsheathed. Other than that, the sword retains the rest of it's pre-snap properties, such as strength, cutting power, and ability to be magically enhanced.

Since "losing" the fight with Theodore, Andrew has stopped minmaxxing her so much and has been exploring more diverse applications for her magic abilities. This broadened focus has diminished her previous abilities, as narrow practice with magic applies mild bonuses and multipliers to those powers. This change in priority is reflected in the paler color of her tools. However, each kind of magic power has a not quite logarithmic curve for improvement/ investment. This means that to anyone but the most astute magic users, Andrew looks just as fast and deadly as she did when using her main skills. In fact, overall she's slightly more effective as she has more diverse tools to deal with any situation.

Andrew doesn't wear any sort of upper clothing at all since her armor and bodysuit were destroyed in the fight. Barring prolonged exposure to extreme circumstances like freezing weather or the vacuum of space, Andrew will never cover her body from the pubes up. Andrew now styles all of her hair into spikes, armpit and pubes included. Andrew's only sort of regular clothing are her shiny black pants, which attach to boots the same way Jotaro's do in JoJo's part 6.

"Losing" to Theodore and a harsh exchange with Jordan afterward prompted Andrew to change her demeanor and goals in life. Without anyone in the solar system who would be a meaningful enemy, much less a genuine threat, Andrew returns to her home city to rebuild in accordance with her new values.

OH YEAH EDIT::: she lost her right eye in the fight too, she’s a cyclops now.

[friends-w-ghosts]

I have an open note Precalc exam soon and finished all my notes, only to realize I have free will and am an artist and there’s nothing stopping me from drawing FFVI characters all over it.

So now I’ve got Terra reminding me what the properties of logarithms are.

[kiskisur]

Lambert W Function, W(x)

In mathematics, the Lambert W function, also called the product logarithm, is a multivalued function, namely the branches of the converse relation of the function f(w) = we^w, where w is any complex number and e^w is the exponential function.

For each integer k there is one branch, denoted by W[k](z), which is a complex-valued function of one complex argument. W[0] is known as the principal branch. These functions have the following property: if z and w are any complex numbers, then

we^w = z, which holds if and only if

w = W[k](z) for some integer k.

When dealing with real numbers only, the two branches W[0] and W[-1] suffice: for real numbers x and y the equation

ye^y = x

can be solved for y only if x ≥ −1/e.

; we get y = W[0](x) if x ≥ 0 and the two values y = W[0](x) and y = W[-1](x) if −1/e ≤ x < 0.

The Lambert W relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance, in the enumeration of trees. It can be used to solve various equations involving exponentials.

For example, The maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions)

It also occurs in the solution of delay differential equations, such as y′(t) = ay(t − 1).

In biochemistry, and in particular enzyme kinetics, an opened-form solution for the time-course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.

mahal is this u..

[darker-than-darkstorm]

... this is amusing. I probably could solve it, but I'd have to dig up some equations somewhere.

The burger is 6, easy. The fries are easy but the concept is difficult: it's the imaginary number i, which doesn't exist and cannot exist (fries x fries has to equal -1 for the second equation to work, and the no real number multiplied by itself can ever equal a negative number. Hence, imaginary numbers, which do have that property)

Now, let's call the glass x to simplify matters, you have x to the power of i, minus x, equals 3. By definition, x to the power of i is equal to e to the power of (i ln x), and by further definition, that equals (cos ln x+i sin ln x).

(ln is the natural logarithm function, e is a transcendental number similar to π with a lot of applications in more complex forms of mathematics, including the natural logarithm function. I won't explain further, but there's a button on most scientific calculators for both of them. There's also buttons for cosine and sine)

So now we're looking at solving the much simpler equation "cos ln x + i sin ln x - x = 3", and ... oh, look, dinner is almost here, so I'm going to stop there. But I'm sure you can easily solve it from there. Good luck!

[alicemark0087]

ACT Math Practice Tips for Mastering Every Section

The ACT Math section can feel like a high-pressure sprint: 60 questions in 60 minutes, covering everything from basic arithmetic to trigonometry. Whether you’re a math whiz or someone who breaks into a sweat at the sight of equations, strategic ACT Math practice is the key to boosting your confidence and score. This guide will walk you through the test’s structure, the topics you need to master, actionable strategies, and the best resources to help you prepare—without any fluff or sales pitches. Let’s dive in!

Understanding the ACT Math Section: What You’re Up Against

The ACT Math test is a 60-minute, 60-question marathon designed to assess skills you’ve learned up to the start of 12th grade. It’s multiple-choice, calculators are allowed (with some restrictions), and there’s no penalty for guessing—so always answer every question! Here’s what you need to know about the content and structure:

Content Breakdown

The test focuses on six core areas, weighted by approximate percentage:

Pre-Algebra (20–25%) involves fractions, ratios, percentages, and basic number operations. These topics form the foundation of many questions on the test.

Elementary Algebra (15–20%) covers solving linear equations, inequalities, and simplifying expressions. A strong grasp of these concepts is essential for handling more complex algebraic problems.

Intermediate Algebra (15–20%): includes quadratic equations, functions, and systems of equations. This section tests your ability to solve more advanced equations and interpret complex algebraic relationships.

Coordinate Geometry (15–20%) focuses on graphing lines, circles, and understanding slopes and distance formulas. Mastering these concepts is key to solving geometry problems on the coordinate plane.

Plane Geometry (20–25%) involves the properties of shapes, angles, and geometric proofs. Understanding these concepts is essential for geometry-based questions on the test.

Trigonometry (5–10%) involves right triangles, sine/cosine/tangent functions, and basic trigonometric identities. While this section is smaller, it's still important to understand these concepts well.

You’ll also receive three subscores (Pre-Algebra/Elementary Algebra, Intermediate Algebra/Coordinate Geometry, and Plane Geometry/Trigonometry), which help pinpoint strengths and weaknesses.

Key Logistics

No formula sheet: You won’t get a formula sheet, so make sure to memorize essential formulas like the quadratic formula and the area of a circle before the test.

Calculator policy: Most graphing calculators are allowed, but avoid models with a computer algebra system (CAS). Double-check your calculator ahead of time to ensure it meets ACT guidelines.

Pacing: Aim for one minute per question. Prioritize easier problems first, quickly solving them and returning to more difficult ones later to maximize your score.

Key Topics to Focus On During ACT Math Practice

While the ACT covers a broad range of math concepts, certain topics appear frequently. Here’s what to prioritize:

Pre-Algebra & Elementary Algebra

These foundational topics make up nearly 40% of the test. Focus on:

Word problems involving ratios, percentages, and proportions, which are often framed in real-life scenarios.

Solving linear equations and inequalities, with an emphasis on real-world contexts such as determining the cost of items after tax or finding the time required for a journey.

Basic statistics, including mean, median, mode, and probability, and their applications in everyday situations like analyzing data or predicting outcomes.

Intermediate Algebra & Coordinate Geometry

These sections test your ability to solve more complex equations and interpret graphs:

Quadratic equations especially through factoring, completing the square, and applying the quadratic formula, which are essential for understanding more advanced mathematical concepts.

Functions including linear, polynomial, and logarithmic types, which are key in analyzing real-world trends such as growth patterns, financial models, and scientific data.

Graphing lines and circles, along with analyzing slopes, midpoints, and distances between points, which will test your spatial reasoning and understanding of coordinate geometry.

Plane Geometry & Trigonometry

Though trigonometry is the smallest category, it’s often the trickiest for students:

Area and volume calculations for two-dimensional and three-dimensional shapes like triangles, circles, spheres, and pyramids.

Understanding triangle properties such as the Pythagorean theorem, and the principles of similar and congruent triangles.

Basic trigonometric ratios such as sine, cosine, and tangent (SOH-CAH-TOA) along with unit circle concepts.

Top Strategies to Maximize Your Score

Knowing the content isn’t enough—you need smart test-taking tactics. Here’s how to practice effectively:

Simulate Real Test Conditions

Taking timed practice tests weekly will help build your stamina and pacing for the ACT. Using official ACT tests provides the most accurate experience and prepares you for the real exam. After each test, review your mistakes thoroughly. Reflect on whether the error was due to a calculation mistake, a misread question, or a gap in your knowledge.

Master Time-Saving Tricks

For algebra, try plugging in answer choices instead of solving from scratch. Eliminate obviously wrong answers to improve guessing odds. Use your calculator only for complex calculations, like trigonometry.

Avoid Common Pitfalls

It’s important not to over-rely on your calculator, as some problems can be solved faster mentally or with scratch work. Always double-check the units and wording of questions, especially if they involve measurements. For example, a question asking for the "radius" but giving the "diameter" is a common trap to watch out for.

The Best Resources for ACT Math Practice

You don’t need to spend a fortune to prepare well. Here are trusted free and paid tools:

Free Resources

Official ACT Practice Tests are the best for realistic questions and can be downloaded from the ACT website. Khan Academy offers free video tutorials on algebra, geometry, and trigonometry. Varsity Tutors provides diagnostic tests and concept-specific drills.

Paid Resources

The Official ACT Prep Guide includes six full-length practice tests with detailed explanations. PrepScholar offers an online course that adapts to your strengths and weaknesses. Barron’s ACT Math Workbook focuses on problem-solving strategies and high-yield topics.

Pro Tip: Build a Study Schedule

Start early with 2–3 months of consistent practice. Mix content review with practice tests, spending about 30% on learning concepts and 70% on applying them. Track progress weekly to note improvements in speed and accuracy.

Final Thoughts: Turning Practice Into Progress

The ACT Math section isn’t about being a human calculator—it’s about strategy, pacing, and knowing where to focus your energy. By targeting high-impact topics, practicing under timed conditions, and using mistakes as learning tools, you’ll build the skills to tackle even the toughest questions. Remember, consistency is key: Even 20–30 minutes of daily practice can lead to significant improvements. Now grab that calculator, hit the books, and get ready to crush this test!