Witches have long been a subject of fascination and mystery. From fairy tales to Hollywood movies, witches have captured our imagination. But what do witches really trust in? First and foremost, witches trust in their own intuition and inner power. They believe in their ability to tap into unseen forces and harness them for their own purposes. Whether it's casting spells, brewing potions, or performing rituals, witches rely on their own instincts and knowledge to create the desired outcome. Secondly, witches trust in the power of nature.
You're probably not going to notice the extra weight of this rope on a short sport climb, but you'll feel it in your pack or on longer routes.
This is great for soft catches on lead, but can be dangerous or even scary for a seconder, particularly if there are obstacles on a climb that you don t want to hit, like a block or slab. In this Perspective article, we propose that a major philosophical change would benefit this field, a proposition that is based on evaluation of situations in which rodent models have provided useful guidance and in which they have led to disappointments.
Secondly, witches trust in the power of nature. Many witches practice nature-based religions and believe in the interconnectedness of all living things. They find comfort and guidance in the cycles of the natural world and often incorporate elements such as herbs, crystals, and the moon into their practices.
Coiling Rope in a Box
What is the longest rope length L of radius r that can fit into a box? The rope is a smooth curve with a tubular neighborhood of radius r, such that the rope does not self-penetrate. For an open curve, each endpoint is surrounded by a ball of radius r. For a box of dimensions $11\frac$ and rope with $r=\frac$, perhaps $L=\frac+\frac<\pi> \approx 1.3$, achieved by a 'U':
I know packing circles in a square is a notoriously difficult problem, but perhaps it is easier to pack a rope in a cube, because the continuity of the curve constrains the options? (I struggle with this every fall, packing a gardening hose in a rectangular tub.) I am more interested in general strategies for how to best coil the rope, rather than specific values of L. It seems that if r is large w.r.t. the box dimensions (as in the above example), no "penny-packing" cross-sectional structure is possible, where one layer nestles in the crevices of the preceding layer.
This is a natural question and surely has been explored, but I didn't find much. Edit 1. It seems a curvature constraint is needed to retain naturalness: The curve should not turn so sharply that the disks of radius r orthogonal to the curve that determine the tubular neighborhood interpenetrate. Edit 2 (26Jun10). See also the MO question concerning decidability. Edit 3 (12Aug10). Here is an observation on the 2D version, where a $1 1 2r$ box may only accommodate one layer of rope. If $k=\frac$ is an even integer, then I can see two natural strategies for coiling the rope within the box:
$\qquad \qquad \qquad \qquad \qquad$ Red is rope core curve, blue marks the rope boundary.
Interestingly, if I have calculated correctly, the length of the red rope curve is identical for the two strategies: $$L = 2 (k-1)[r \pi/2] + 2(k-1)^2 r \;. $$ For $r=\frac$, $k=8$ as illustrated, $L=\frac<7\pi> + \frac \approx 7.5$. (As a check, for $r=\frac$, $k=2$, and $L$ evaluates to $\frac<\pi>+\frac$ as in the first example above.)
- packing-and-covering
- mg.metric-geometry
- dg.differential-geometry
- open-problems
Witches also trust in the power of intention and visualization. They understand that thoughts and emotions have the power to manifest in the physical world. By focusing their thoughts and intentions on what they desire, witches believe they can bring about change and shape their reality. Furthermore, witches trust in the power of community and collective energy. Covens and gatherings are common among witches, providing a sense of support, validation, and shared knowledge. They believe that coming together with others who share their beliefs and practices amplifies their power and can bring about greater results. Finally, witches trust in the concept of balance and karma. They recognize that every action has a consequence and seek to use their powers responsibly and with harm to none. This belief in karma reinforces their commitment to ethical practices and responsible magic. In conclusion, witches trust in their own intuition, the power of nature, intention and visualization, community and collective energy, and the concept of balance and karma. Through these beliefs, witches find empowerment, guidance, and a deep connection to the unseen forces of the universe..
Reviews for "Embracing the Shadows: What Witches Trust in Dark Magic"
1. Sarah - 1/5 stars - "I was really disappointed with 'What do witches trust in'. The plot was confusing and jumbled, and I struggled to follow along with what was happening. The characters were flat and lacked depth, making it hard for me to care about their journey. The writing style itself was also quite clunky and awkward. Overall, I found this book to be a frustrating read and I would not recommend it."
2. John - 2/5 stars - "I had high hopes for 'What do witches trust in' but unfortunately, it fell short of my expectations. The story started off promising but quickly became convoluted and hard to follow. The dialogue felt forced and unnatural, making it difficult for me to connect with the characters. Additionally, I found the pacing to be inconsistent and the ending felt rushed. While there were some interesting concepts, they were not executed well. This book had potential, but it ultimately left me feeling disappointed."
3. Emma - 2/5 stars - "I struggled to get through 'What do witches trust in'. The writing felt disjointed and lacked cohesion, which made it hard for me to stay engaged with the story. The characters were one-dimensional and their motivations were unclear. The plot didn't seem well thought out and had many holes. I also found the pacing to be off, with long stretches of boredom followed by rushed action. Overall, I was unimpressed with this book and would not recommend it to others."